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RELATIVITY

FHE SPECIAL " THE GENERAL THEORY

A P PULAR EXPOSITION

BY

ALBERT EINSTE-IN, Ph.D.PROFESSOR OF PHYSICS IN THB UNIVERSITY OF BERLIN

AUTHORISED TRANSLATION BY

ROBERT W. LAWSON, D.Sc.

UNIVERSITY OF SHEFFIELD

WITH FIVE DIAGRAMS

AND A PORTRAIT OF THE AUTHOR

THIRD EDITION

METHUEN " GO. LTD.

36 ESSEX STREET W.G.

LONDON

CT

This Translationwas

Jirst Published. .

August iqthrqso

Second Edition September 1920

Third Editioniqzo

PREFACE

THE present book is intended, as far as possible,

to give an exact insight into the theory of Re

lativity to those readers who, from a general

scientific and philosophical point of view, are interested

in the theory, but who are not conversant with the

mathematical apparatus l of theoretical physics. The

work presumes a standard of education corresponding

to that of a university matriculation examination,

and, despite the shortness of the book, a fair amount

of patience and force of will on the part of the reader.

The author has spared himself no pains in his endeavour

1 The mathematical fundaments of the special theory of

relativity are to be found in the original papers of H. A. Lorentz,

A. Einstein, H. Minkowski, published under the title Das

Relativitdtsprinzip (The Principle of Relativity) in B. G.

Teubner's collection of monographs Fortschritte der mathe-

matischen Wissenschajten (Advances in the Mathematical

Sciences), also in M. Laue's exhaustive book Das Relativitdts

prinzip " published by Friedr. Vieweg " Son, Braunschweig.

The general theory of relativity, together with the necessary

parts of the theory of invariants, is dealt with in the author's

book Die Grundlagen der allgcmeinen Relativitdtstheorie (The

Foundations of the General Theory of Relativity) " Joh. Ambr.

Barth, 1916 ; this book assumes some familiarity with the special

theory of relativity.

vi RELATIVITY

to present the main ideas in the simplestand most in

telligibleform, and on the whole, in the sequence and con

nection in which theyactuallyoriginated.In the interest

of clearness,it appeared to me inevitable that I should

repeat myself frequently,without paying the slightest

attention to the eleganceof the presentation. I adhered

scrupulouslyto the precept of that brilliant theoretical

physicistL. Boltzmann, accordingto whom matters of

eleganceought to be left to the tailor and to the cobbler.

I make no pretence of having withheld from the reader

difficulties which are inherent to the subject. On the

other hand, I have purposely treated the empirical

physicalfoundations of the theoryin a"

step-motherly"

fashion, so that readers unfamiliar with physicsmay

not feel like the wanderer who was unable to see the

forest for trees. May the book bring some one a few

happy hours of suggestivethought !

December, 1916 A. EINSTEIN

NOTE TO THE THIRD EDITION

INthe present year (1918)an excellent and detailed

manual^on the generaltheory of relativity,written

by H. Weyl, was published by the firm Julius

Springer (Berlin).This book, entitled Raum " Zsit "

Materie (Space" Time " Matter),may be warmly recom

mended to mathematicians and physicists.

BIOGRAPHICAL NOTE

ALBERTEINSTEIN is the son of German-

Jewish parents. He was born in 1879 m tne

town of Ulm, Wurtemberg, Germany. His

schooldays were spent in Munich, where he attended

the Gymnasium until his sixteenth year. After leaving

school at Munich, he accompanied his parents to Milan,

whence he proceeded to Switzerland six months later

to continue his studies.

From 1856 to 1900 Albert Einstein studied mathe

matics and physics at the Technical High School in

Zurich, as he intended becoming a secondary school

(Gymnasium) teacher. For some time afterwards he

was a private tutor, and having meanwhile become

naturalised, he obtained a post as engineer in the Swiss

Patent Office in 190^ which position he occupied till

1Z--7

1900. The main ideas involved in the most important

of Einstein's theories date back to this period. Amongst

these may be mentioned : The Special Theory of Rela

tivity, Inertia of Energy, Theory of the Brownian Move

ment, and the Quantum-Law of the Emission and Ab

sorption of Light (1905).

These were followed some years

viii RELATIVITY

later by the Theory of the SpecificHeat of Solid Bodies,

and the fundamental idea of the General Theory of

Relativity. -XQ 3~V"During the interval 1909 to 1911 he occupied the post

of Professor Extraor dinar ius at the Universityof Zurich,

afterwards being appointed to the Universityof Prague,

Bohemia, where he remained as Professor Ordinarius

until 1912. In the latter year Professor Einstein

accepted a similar chair at the Polytechnikum,Zurich,

and continued his activities there until 1914, when he

received a call to the Prussian Academy of Science,

Berlin, as successor to Van't Hoff. Professor Einstein

is able to devote himself freelyto his studies at the

Berlin Academy, and it was here that he succeeded in

completing his work on the General Theory of Relativity

(1915-17). Professor Einstein also lectures on various

specialbranches of physics at the Universityof Berlin,

and, in addition, he is Director of the Institute for

Physical Research of the Kaiser Wilhelm Gesellschaft.

Professor Einstein has been twice married. His first

wife, whom he married at Berne in 10,03, was a fellow-

student from Serbia. There were two sons of this

marriage, both of whom are living in Zurich, the elder

being sixteen years of age. Recently Professor Einstein

married a widowed cousin, with whom he is now living

in Berlin.

R. W. L.

TRANSLATOR'S NOTE

IN presenting this translation to the English-

reading public, it is hardly necessary for me to

enlarge on the Author's prefatory remarks, except

to draw attention to those additions to the book which

do not appear in the original.

At my request, Professor Einstein kindly supplied

me with a portrait of himself, by one of Germany's

most celebrated artists. Appendix III, on" The

Experimental Confirmation of the General Theory of

Relativity," has been written specially for this trans

lation. Apart from these valuable additions to the book,

I have included a biographical note on the Author,

and, at the end of the book, an Index and a list of

English references to the sub ject.

This list,

which is more

suggestive than exhaustive, is intended as a guide to those

readers who wish to pursue the subject farther.

I desire to tender my best thanks to my colleagues

Professor S. R. Milner, D.Sc., and Mr. W. E. Curtis,

A.R.C.SC., F.R.A.S., also to my friend Dr. Arthur

Holmes, A.R.C.Sc., F.G.S., of the Imperial College,

for their kindness in reading through the manuscript,

xRELATIVITY

for helpful criticism, and for numerous suggestions. I

owe an expression of thanks also to Messrs. Methuen

for their ready counsel and advice, and for the care

they have bestowed on the work during the course of

its publication.

ROBERT W. LAWSON

THE PHYSICS LABORATORY

THE UNIVERSITY OF SHEFFIELD

June 12, 1920

CONTENTS

PART I

THE SPECIAL THEORY OF RELATIVITY

PAGE

I. Physical Meaning of Geometrical Propositions.

i

II. The System of Co-ordinates. " "

5

UK. Space and Time in Classical Mechanics. . 9

IV. The Galileian System of Co-ordinates.

.11

V. The Principle of Relativity (in the Restricted

Sense). . . . .

.12

VI. The Theorem of the Addition of Velocities em

ployed in Classical Mechanics. .

16

VII. The Apparent Incompatibility of the Law of

Propagation of Light with the Principle of

Relativity. . . .

17

'III. On the Idea of Time in Physics. .

.21

IX. The Relativity of Simultaneity. .

.25

X. On the Relativity of the Conception of Distance 28

XI. The Lorentz Transformation. .

.30

XII. The Behaviour of Measuring -Rods and Clocks

in Motion. . . . -35

Xll

XIII. Theorem of the Addition of Velocities. The

Experiment of Fizeau. . -3

XIV. The Heuristic Value of the Theory of Relativity 4

XV. General Results of the Theory . . .4,

XVI. Experience and the SpecialTheory of Relativity 4'

XVII. Minkowski's Four-dimensional Space . . "5;

PART II

THE GENERAL THEORY OF RELATIVITY

XVIII. Specialand General Principleof Relativity . 5

XIX. The Gravitational Field. . .

.6

XX. The Equalityof Inertial and Gravitational Mass

as an Argument for the General Postulate

of Relativity .....

XXI. In what Respects are the Foundations of Clas

sical Mechanics and of the SpecialTheoryof Relativityunsatisfactory? .

XXII. A Few Inferences from the General Principleof

Relativity .....

XXIII. Behaviour of Clocks and Measuring-Rods on a

Rotating Body of Reference.

XXIV. Euclidean and Non-Euclidean Continuum

XXV. Gaussian Co-ordinates....

XXVI. The Space-time Continuum of the SpecialTheory of Relativityconsidered as a

Euclidean Continuum

CONTENTS xiii

PAGE

XXVII. The Space-time Continuum of the General

Theory of Relativityis not a Euclidean

Continuum. . . . -93

XXVIII. Exact Formulation of the General Principleof

Relativity . . . . -97

XXIX. The Solution of the Problem of Gravitation on

the Basis of the General Principle of

Relativity .....too

PART III

CONSIDERATIONS ON THE UNIVERSE

AS A WHOLE

XXX. Cosmological Difficulties of Newton's Theory 105

XXXI. The Possibilityof a "Finite" and yet "Un

bounded" Universe.. . .

108

XXXII. The Structure of Space according to the

General Theory of Relativity . . 113

APPENDICES

I. Simple Derivation of the Lorentz Transformation. 115

II. Minkowski's Four-dimensional Space ("World")

[Supplementaryto Section XVII.]. .

121

III. The Experimental Confirmation of the General

Theory of Relativity. . ., .123

(a) Motion of the Perihelion of Mercury . 124

(b) Deflection of Light by a Gravitational Field 126

(c) Displacement of SpectralLines towards the

Red. . . . . 129

BIBLIOGRAPHY. . . . . . 133

. . . . . . .135

RELATIVITY

THE SPECIAL AND THEGENERAL

THEORY

RELATIVITY

PART I

THE SPECIAL THEORY OF RELATIVITY

PHYSICAL MEANING OF GEOMETRICAL

PROPOSITIONS

IN your schooldays most of you who read this

book made acquaintance with the noble building of

Euclid's geometry, and you remember" perhaps

with more respect than love"

the magnificent structure,

on the lofty staircase of which you were chased about

for uncounted hours by conscientious teachers. By

reason of your past experience, you would certainly

regard everyone with disdain who should pronounce even

the most out-of-the-way proposition of this science to

be untrue. But perhaps this feeling of proud certainty

would leave you immediately if some one were to ask

you :"

What, then, do you mean by the assertion that

these propositions are true ?" Let us proceed to give

this question a little consideration.

Geometry sets out from certain conceptions such as

"

plane,"" point," and

"

straight line," with which

i

2 SPECIAL THEORY OF RELATIVITY

we are able to associate more or less definite ideas, and

from certain simple propositions(axioms) which,

in virtue of these ideas, we are inclined to accept as

" true." Then, on the basis of a logicalprocess, the

justificationof which we feel ourselves compelled to

admit, all remaining propositionsare shown to follow

from those axioms, i.e.they are proven. A propositionis then correct ("true ") when it has been derived in the

recognisedmanner from the axioms. The questionof the

" truth" of the individual geometricalproposi

tions is thus reduced to one of the " truth " of the

axioms. Now it has long been known that the last

questionis not only unanswerable by the methods of

geometry, but that it is in itselfentirelywithout mean

ing. We cannot ask whether it is true that only one

straightline goes through two points. We can only

say that Euclidean geometry deals with thingscalled" straightlines,"to each of which is ascribed the pro

perty of being uniquely determined by two pointssituated on it. The concept

"

true"

does not tallywith

the assertions of pure geometry, because by the word"

true"

we are eventuallyin the habit of designatingalways the correspondence with a

" real "

object;

geometry, however, is not concerned with the relation

of the ideas involved in it to objectsof experience,but

only with the logicalconnection of these ideas among

themselves.

It is not difficultto understand why, in spiteof this,

we feel constrained to call the propositionsof geometry" true." Geometrical ideas correspond to more or less

exact objectsin nature, and these last are undoubtedlythe exclusive cause of the genesisof those ideas. Geo

metry ought to refrain from such a course, in order to

GEOMETRICAL PROPOSITIONS 3

give to its structure the largestpossiblelogicalunity.The practice,for example, of seeingin a

" distance "

two marked positionson a practicallyrigidbody is

something which islodgeddeeplyin our habit of thought.We are accustomed further to regard three pointsas

being situated on a straightline, if their apparent

positionscan be made to coincide for observation with

one eye, under suitable choice of our placeof observa

tion.

If, in pursuance of our habit of thought, we now

supplement the propositionsof Euclidean geometry bythe singlepropositionthat two points on a practically

rigidbody always correspond to the same distance

(line-interval),independentlyof any changes in position

to which we may subjectthe body, the propositionsof

Euclidean geometry then resolve themselves into pro

positionson the possiblerelative positionof practically

rigidbodies.1 Geometry which has been supplementedin this way is then to be treated as a branch of physics.We can now legitimatelyask as to the "

truth "

of

geometricalpropositionsinterpretedin this way, since

we are justifiedin asking whether these propositions

are satisfied for those real things we have associated

with the geometricalideas. In less exact terms we can

express this by saying that by the "

truth "

of a geo

metrical propositionin this sense we understand its

validityfor a construction with ruler and compasses.

1 It follows that a natural object is associated also with a

straightline. Three points A, B and C on a rigidbody thus

lie in a straightline when, the points A and C being given, B

is chosen such that the sum of the distances AD and BC is as

short as possible. This incomplete suggestion will suffice for

our presentpurpose.


Of course the conviction of the " truth "

ofgeo

metrical propositions in this sense is founded exclusively

onrather incomplete experience. For the present we

shall assume the"

truth"

of the geometrical proposi

tions, then at a later stage (in the general theory of

relativity) we shall see that this " truth " is limited,

and we shall consider the extent of its limitation.

= ns

ii

THE SYSTEM OF CO-ORDINATES

ONthe basis of the physical interpretation of dis

tance which has been indicated, we are also

in a position to establish the distance between

two points on a rigid body by means of measurements.

For this purpose we require a" distance "

(rod 5)

which is to be used once and for all, and which we

employ as a standard measure. If, now, A and B are

two points on a rigid body, we can construct the

line joining them according to the rules of geometry ;

then, starting from A, we can mark off the distance

S time after time until we reach B. The number of

these operations required is the numerical measure

of the distance AB. This is the basis of all measure

ment of length.1

Every description of the scene of an event or of the

position of an object in space is based on the specifica

tion of the point on a rigid body (body of reference)

with which that event or object coincides. This applies

not only to scientific description, but also to everyday

life. If I analyse the place specification"

Trafalgar

1 Here we have assumed that there is nothing left over, i.e.

that the measurement gives a whole number. This difficultyis got over by the use of divided measuring-rods, the introduction

of which does not demand any fundamentally new method.


Square, London,"1 I arrive at the followingresult.

The earth is the rigidbody to which the specification

of place refers ;" Trafalgar Square, London," is a

well-defined point,to which a name has been assigned,and with which the event coincides in space.2

This primitive method of place specificationdeals

only with placeson the surface of rigidbodies, and is

dependent on the existence of points on this surface

which are distinguishablefrom each other. But we

can free ourselves from both of these limitations without

alteringthe nature of our specificationof position.If, for instance, a cloud is hovering over Trafalgar

Square, then we can determine its positionrelative to

the surface of the earth by erectinga pole perpendicu

larlyon the Square, so that it reaches the cloud. The

lengthof the polemeasured with the standard measuring-

rod, combined with the specificationof the positionof

the foot of the pole,suppliesus with a complete placespecification.On the basis of this illustration,we are

able to see the manner in which a refinement of the con

ception of positionhas been developed.

(a)We imagine the rigidbody, to which the place

specificationis referred,supplemented in such a manner

that the objectwhose positionwe requireis reached bythe completed rigidbody.

(b]In locatingthe positionof the object,we make

use of a number (herethe length of the polemeasured

1 I have chosen this as being more familiar to the Englishreader than the " Potsdamer Platz, Berlin," which is referred to

in the original. (R. W. L.)2 It is not necessary here to investigatefurther the significance

of the expression " coincidence in space." This conception is

sufficientlyobvious to ensure that differences of opinion ar*

scarcelylikelyto arise as to its applicabilityin practice.

THE SYSTEM OF CO-ORDINATES 7

with the measuring-rod)instead of designatedpointsof

reference.

(c)We speak of the heightof the cloud even when the

pole which reaches the cloud has not been erected.

By means of opticalobservations of the cloud from

different positionson the ground, and takinginto account

the propertiesof the propagationof light,we determine

the lengthof the pole we should have requiredin order

to reach the cloud.

From this consideration we see that it will be ad

vantageous if,in the descriptionof position,itshould be

possibleby means of numerical measures to make our

selves independentof the existence of marked positions

(possessingnames) on the rigidbody of reference. In

the physics of measurement this is attained by the

applicationof the Cartesian system of co-ordinates.

This consists of three plane surfaces perpendicularto each other and rigidlyattached to a rigidbody.Referred to a system of co-ordinates,the scene of any

event will be determined (forthe main part)by the

specificationof the lengthsof the three perpendicularsor co-ordinates (x,y, z)which can be dropped from the

scene of the event to those three plane surfaces. The

lengths of these three perpendicularscan be deter

mined by a series of manipulationswith rigidmeasuring-rods performed accordingto the rules and methods laid

down by Euclidean geometry.In practice,the rigidsurfaces which constitute the

system of co-ordinates are generally not available ;

furthermore, the magnitudes of the co-ordinates are not

actuallydetermined by constructions with rigidrods,but

by indirect means. If the results of physicsand astron

omy are to maintain their clearness,the physicalmean-


ing of specifications of position must always be sought

in accordance with the above considerations.1

We thus obtain the following result : Every descrip

tion of events inspace

involves the use of a rigid body

to which such events have to be referred. The resulting

relationship takes for granted that the laws of Euclidean

geometry hold for" distances," the " distance "

being

represented physically by means of the convention of

two marks on a rigid body.

1 A refinement and modification of these views does not become

necessaryuntil we come to deal with the general theory of

relativity, treated in the second part of this book.

Ill

SPACE AND TIME IN CLASSICAL MECHANICS

THE purpose of mechanics is to describe how

bodies change their position in space with

time." I should load my conscience with grave

sins against the sacred spirit of lucidity were I to

formulate the aims of mechanics in this way, without

serious reflection and detailed explanations. Let us

proceed to disclose these sins.

It is not clear what is to be understood here by"

position"

and"

space." I stand at the window of a

railway carriage which is travelling uniformly, and drop

a stone on the embankment, without throwing it. Then,

disregarding the influence of the air resistance, I see the

stone descend in a straight line. A pedestrian who

observes the misdeed from the footpath notices that the

stone falls to earth in a parabolic curve. I now ask :

Do the "

positions"

traversed by the stone lie " in

reality"

on a straight line or on a parabola ? Moreover,

what is meant here by motion "

in space" ? From the

considerations of the previous section the answer is

self-evident. In the first place, we entirely shun the

vague word "

space," of which, we must honestly

acknowledge, we cannot form the slightest concep

tion, and we replace it by" motion relative to a

practically rigid body of reference." The positions

relative to the body of reference (railway carriage or

embankment) have already been defined in detail in the


precedingsection. If instead of "

body of reference "

we insert"

system of co-ordinates,"which is a useful

idea for mathematical description,we are in a position

to say : The stone traverses a straightline relative to a

system of co-ordinates rigidlyattached to the carriage,but relative to a system of co-ordinates rigidlyattached

to the ground (embankment) it describes a parabola.With the aid of this example it is clearlyseen that there

is no such thing as an independentlyexistingtrajectory

(lit."

path-curve"

1),but only a trajectoryrelative to a

particularbody of reference.

In order to have a completedescriptionof the motion,

we must specifyhow the body alters its positionwith

time ; i.e.for every point on the trajectoryit must be

stated at what time the body is situated there. These

data must be supplemented by such a definition of

time that, in virtue of this definition,these time-values

can be regarded essentiallyas magnitudes (resultsof

measurements) capable of observation. If we take our

stand on the ground of classical mechanics, we can

satisfythis requirement for our illustration in the

followingmanner. We imagine two clocks of identical

construction ; the man at the railway-carriagewindowis holding one of them, and the man on the foot

path the other. Each of the observers determines

the positionon his own reference-bodyoccupied by the

stone at each tick of the clock he is holding in his

hand. In this connection we have not taken account

of the inaccuracyinvolved by the finiteness of the

velocityof propagationof light. With this and with a

second difficultyprevailinghere we shall have to deal

in detail later.

1 That is,a curve along which the body moves.

IV

THE GALILEIAN SYSTEM OF CO-ORDINATES

ASis well known, the fundamental law of the

mechanics of Galilei-Newton, which is known

as the law of inertia, can be stated thus :

A body removed sufficiently far from other bodies

continues in a state of rest or of uniform motion

in a straight line. This law not only says some

thing about the motion of the bodies, but it also

indicates the reference -bodies or systems of co

ordinates, permissible in mechanics, which can be used

in mechanical description. The visible fixed stars are

bodies for which the law of inertia certainly holds to a

high degree of approximation. Now if we use a system

of co-ordinates which is rigidly attached to the earth,

then, relative to this system, every fixed star describes

a circle of immense radius in the course of an astrono

mical day, a result which is opposed to the statement

of the law of inertia. So that if we adhere to this law

we must refer these motions only to systems of co

ordinates relative to which the fixed stars do not move

in a circle. A system of co-ordinates of which the state

of motion is such that the law of inertia holds relative to

it is called a" Galileian system of co-ordinates." The

laws of the mechanics of Galilei-Newton can be regarded

as valid only for a Galileian system of co-ordinates.

THE PRINCIPLE OF RELATIVITY (IN THE

RESTRICTED SENSE)

INorder to attain the greatest possible clearness,

let us return to our example of the railway carriage

supposed to be travelling uniformly. We call its

motion a uniform translation (" uniform " because

it is of constant velocity and direction, " translation "

because although the carriage changes its position

relative to the embankment yet it does not rotate

in so doing). Let us imagine a raven flying through

the air in such a manner that its motion, as observed

from the embankment, is uniform and in a straight

line. If we were to observe the flying raven from

the moving railway carriage, we should find that the

motion of the raven would be one of different velo

city and direction, but that it would still be uniform

and in a straight line. Expressed in an abstract

manner we may say : If a mass m is moving uni

formly in a straight line with respect to a co-ordinate

system K, then it will also be moving uniformly and in a

straight line relative to a second co-ordinate system

K', provided that the latter is executing a uniform

translatory motion with respect to K. In accordance

with the discussion contained in the preceding section,

it follows that :

If K is a Galileian co-ordinate system, then every other

co-ordinate system K' is a Galileian one, when, in rela

tion to K, it is in a condition of uniform motion of trans

lation. Relative to K' the mechanical laws of Galilei-

Newton hold good exactly as they do with respect to K.

We advance a step farther in our generalisationwhen

we express the tenet thus : If, relative to K, K' is a

uniformlymoving co-ordinate system devoid of rotation,

then natural phenomena run their course with respect to

K' according to exactlythe same general laws as with

respect to K. This statement is called the principle

of relativity(inthe restricted sense).

As long as one was convinced that all natural pheno

mena were capable of representationwith the help of

classical mechanics, there was no need to doubt the

validityof this principleof relativity.But in view of

the more recent development of electrodynamics and

opticsit became more and more evident that classical

mechanics affords an insufficient foundation for the

physicaldescriptionof all natural phenomena. At this

juncturethe question of the validityof the principleof

relativitybecame ripe for discussion, and it did not

appear impossible that the answer to this question

might be in the negative.

Nevertheless, there are two general facts which at the

outset speak very much in favour of the validityof the

principleof relativity.Even though classical mechanics

does not supply us with a sufficientlybroad basis for the

theoretical presentation of all physical phenomena,stillwe must grant it a considerable measure of " truth,"

since it supplies us with the actual motions of the

heavenly bodies with a delicacyof detail little short of

wonderful. The principleof relativitymust therefore

apply with great accuracy in the domain of mechanics.

But that a principleof such broad generalityshould

hold with such exactness in one domain of phenomena,

and yet should be invalid for another, is a priorinot

very probable.We now proceed to the second argument, to which,

moreover, we shall return later. If the principleof rela

tivity(inthe restricted sense) does not hold, then the

Galileian co-ordinate systems K, K', K", etc., which are

moving uniformly relative to each other, will not be

equivalentfor the description of natural phenomena.

In this case we should be constrained to believe that

natural laws are capable of being formulated in a par

ticularlysimple manner, and of course only on condition

that, from amongst all possibleGalileian co-ordinate

systems, we should have chosen one (K0) of a particular

state of motion as our body of reference. We should

then be justified(becauseof its merits for the descriptionof natural phenomena) in callingthis system

"

absolutely

at rest,"and all other Galileian systems K " in motion."

If, for instance, our embankment were the system K^,then our railway carriage would be a system K,

relative to which less simple laws would hold than with

respect to K0. This diminished simplicity would be

due to the fact that the carriage K would be in motion

(i.e." really") with respect to K0. In the general laws

of nature which have been formulated with refer

ence to K, the magnitude and direction of the velocityof the carriagewould necessarilyplay a part. We should

expect, for instance,that the note emitted by an organ-

pipe placed with its axis parallelto the direction of

travel would be different from that emitted if the axis

of the pipe were placed perpendicular to this direction.

THE PRINCIPLE OF RELATIVITY 15

Now in virtue of its motion in an orbit round thesun,

our earth is comparable with a railway carriage travel

ling with a velocity of about 30 kilometresper second.

If the principle of relativity were not valid we should

therefore expect that the direction of motion of the

earth at any moment would enter into the laws of nature,

and also that physical systems in their behaviour would

be dependent on the orientation inspace

with respect

to the earth. For owing to the alteration in direction

of the velocity of revolution of the earth in the course

of a year, the earth cannot be at rest relative to the

hypothetical system K0 throughout the wholeyear.

However, the most careful observations have never

revealed such anisotropic properties in terrestrial physi

cal space,i.e.

a physical non-equivalence of different

directions. This isvery powerful argument in favour

of the principle of relativity.

VI

THE THEOREM OF THE ADDITION OF VELOCI

TIES EMPLOYED IN CLASSICAL MECHANICS

LET us suppose our old friend the railway carriage

to be travelling along the rails with a constant

velocity v, and that a man traverses the length of

the carriage in the direction of travel with a velocity w.

How quickly or, in other words, with what velocity W

does the man advance relative to the embankment

during the process ? The only possible answer seems to

result from the following consideration : If the man were

to stand still for a second, he would advance relative to

the embankment through a distance v equal numerically

to the velocity of the carriage. As a consequence of

his walking, however, he traverses an additional distance

w relative to the carriage, and hence also relative to the

embankment, in this second, the distance w being

numerically equal to the velocity with which he is

walking. Thus in total he covers the distance W=v-\-w

relative to the embankment in the second considered.

We shall see later that this result, which expresses

the theorem of the addition of velocities employed in

classical mechanics, cannot be maintained ; in other

words, the law that we have just written down does not

hold in reality. For the time being, however, we shall

assume its correctness.

16

VII

THE APPARENT INCOMPATIBILITY OF THE

LAW OF PROPAGATION OF LIGHT WITH

THE PRINCIPLE OF RELATIVITY

THERE is hardly a simpler law in physics than

that according to which light is propagated in

empty space. Every child at school knows, or

believes he knows, that this propagation takes place

in straight lines with a velocity 0 = 300,000 km. /sec.

At all events we know with great exactness that this

velocity is the same for all colours, because if this were

not the case, the minimum of emission would not be

observed simultaneously for different colours during

the eclipse of a fixed star by its dark neighbour. By

means of similar considerations based on observa

tions of double stars, the Dutch astronomer De Sitter

was also able to show that the velocity of propaga

tion of light cannot depend on the velocity of motion

of the body emitting the light. The assumption that

this velocity of propagation is dependent on the direc

tion " in space" is in itself improbable.

In short, let us assume that the simple law of the

constancy of the velocity of light c (in vacuum) is

justifiably believed by the child at school. Who would

imagine that this simple law has plunged the con

scientiously thoughtful physicist into the greatest


intellectual difficulties? Let us consider how these

difficultiesarise.

Of course we must refer the process of the propaga

tion of light(and indeed every other process)to a rigid

reference-body(co-ordinatesystem). As such a system

let us againchoose our embankment. We shall imagine

the air above it to have been removed. If a ray of

lightbe sent along the embankment, we see from the

above that the tip of the ray will be transmitted with

the velocityc relative to the embankment. Now let

us suppose that our railwaycarriageis again travelling

along the railwaylines with the velocityv, and that

its direction is the same as that of the ray of light,but

its velocityof course much less. Let us inquireabout

the velocityof propagation of the ray of lightrelative

to the carriage. It is obvious that we can here apply the

consideration of the previoussection, since the ray of

lightplaysthe part of the man walkingalong relativelyto the carriage. The velocityW of the man relative

to the embankment is here replacedby the velocityof lightrelative to the embankment, w is the required

velocityof lightwith respect to the carriage,and we

have

w " c - v.

The velocityof propagation of a ray of lightrelative to

the carriagethus comes out smaller than c.

But this result comes into conflict with the principleof relativityset forth in Section V. For, like everyother general law of nature, the law of the transmission

of lightin vacuo must, accordingto the principleof

relativity,be the same for the railway carriage as1

reference-bodyas when the rails are the body of refeil

THE PROPAGATION OF LIGHT 19

mce. But, from our above consideration, this would

ippear to be impossible. If every ray of lightis pro

pagated relative to the embankment with the velocity

;,then for this reason it would appear that another law

)ipropagationof lightmust necessarilyhold with respect

;o the carriage" a result contradictoryto the principle:"frelativity.

In view of this dilemma there appears to be nothingjlse for it than to abandon either the principleof rela-

[ivityor the simple law of the propagation of lightin

MCUO. Those of you who have carefullyfollowed the

precedingdiscussion are almost sure to expect that

sve should retain the principleof relativity,which

ippealsso convincinglyto the intellect because it is so

natural and simple. The law of the propagation of

iightin vacua would then have to be replaced by a

more complicated law conformable to the principleof

relativity.The development of theoretical physicsshows, however, that we cannot pursue this course.

The epoch-making theoretical investigationsof H. A.

Lorentz on the electrodynamicaland opticalphenomenaconnected with moving bodies show that experiencein this domain leads conclusivelyto a theory of electro

magnetic phenomena, of which the law of the constancyof the velocityof lightin vacua is a necessary conse

quence. Prominent theoretical physicistswere there

fore more inclined to rejectthe principleof relativity,in spiteof the fact that no empiricaldata had been

found which were contradictoryto this principle.At this juncturethe theory of relativityentered the

arena. As a result of an analysisof the physicalcon

ceptionsof time and space, it became evident that in

realitythere is not the least incompatibilityietween the


principle of relativity and the law of propagation of light,

and that by systematically holding fast to both these

lawsa logically rigid theory could be arrived at. This

theory has been called the special theory of relativity

to distinguish it from the extended theory, with which

we shall deal later. In the following pages we shall

present the fundamental ideas of the special theory of

relativity.

VIII

ON THE IDEA OF TIME IN PHYSICS

LIGHTNINGhas struck the rails on our railway

embankment at two places A and B far distant

from each other. I make the additional assertion

that these two lightning flashes occurred simultaneously.

If I ask you whether there is sense in this statement,

you will answer my question with a decided

" Yes." But if I now approach you with the request

to explain to me the sense of the statement more

precisely, you find after some consideration that the

answer to this question is not so easy as it appears at

first sight.

After some time perhaps the following answer would

occur to you :" The significance of the statement is

clear in itself and needs no further explanation ; of

course it would require some consideration if I were to

be commissioned to determine by observations whether

in the actual case the two events took place simul

taneously or not." I cannot be satisfied with this answer

for the following reason. Supposing that as a result

of ingenious considerations an able meteorologist were

to discover that the lightning must always strike the

places A and B simultaneously, then we should be faced

with the task of testing whether or not this theoretical

result is in accordance with the reality. We encounter

the same difficultywith all physicalstatements in which

the conception" simultaneous "

plays a part. The

concept does not exist for the physicistuntil he has the

possibilityof discovering whether or not it is fulfilled

in an actual case. We thus require a definition of

simultaneity such that this definition supplies us with

the method by means of which, in the present case, he

can decide by experiment whether or not both the

lightning strokes occurred simultaneously. As long

as this requirement is not satisfied,I allow myself to be

deceived as a physicist(and of course the same applies

if I am not a physicist),when I imagine that I am able

to attach a meaning to the statement of simultaneity.

(I would ask the reader not to proceed farther until he

is fullyconvinced on this point.)

After thinking the matter over for some time you

then offer the following suggestion with which to test

simultaneity. By measuring along the rails, the

connecting line AB should be measured up and an

observer placed at the mid-point M of the distance AB.

This observer should be supplied with an arrangement

(e.g.two mirrors inclined at 90") which allows him

visuallyto observe both places A and B at the same

time. If the observer perceives the two flashes of

lightningat the same time, then they are simultaneous.

I am very pleased with this suggestion, but for all

that I cannot regard the matter as quitesettled,because

I feel constrained to raise the following objection:

" Your definition would certainly be right,if I onlyknew that the light by means of which the observer

at M perceives the lightningflashes travels along the

length A " " M with the same velocity as along the

length B " " M. But an examination of this supposi-

IDEA OF TIME IN PHYSICS 23

tion would only be possibleif we alreadyhad at our

disposalthe means of measuring time. It would thus

appear as though we were moving here in a logicalcircle."' After further consideration you cast a somewhat

disdainful glance at me " and rightlyso " and you

declare : "I maintain my previousdefinition neverthe

less,because in realityit assumes absolutelynothingabout light. There is only one demand to be made of

the definition of simultaneity,namely, that in every

real case it must supply us with an empiricaldecision

as to whether or not the conception that has to

be defined is fulfilled. That my definition satisfies

this demand is indisputable. That lightrequiresthe

same time to traverse the path A " " M as for the pathB " "M isin realityneither a suppositionnor a hypothesisabout the physicalnature of light,but a stipulationwhich I can make of my own freewill in order to arrive

at a definition of simultaneity."It is clear that this definition can be used to givean

exact meaning not only to two events, but to as many

events as we care to choose, and independently of the

positionsof the scenes of the events with respectto the

body of reference1 (here the railway embankment).

We are thus led also to a definition of " time"

in physics.For this purpose we suppose that clocks of identical

construction are placed at the pointsA, B and C of

1 We suppose further, that, when three events A, B and C

occur in different places in such a manner that A is simul

taneous with B, and B is simultaneous with C (simultaneous

in the sense of the above definition),then the criterion for the

simultaneityof the pair of events A, C is also satisfied. This

assumption is a physicalhypothesisabout the law of propagationof light; it must certainlybe fulfilled if we are to maintain the

law of the constancy of the velocityof lightin vacua.


the railway line (co-ordinate system), and that they

are set in such a manner that the positions of their

pointers are simultaneously (in the above sense) the

same. Under these conditions we understand by the

" time" of an event the reading (position of the hands)

of that one of these clocks which is in the immediate

vicinity (in space) of the event. In this manner a

time-value is associated withevery event which is

essentially capable of observation.

This stipulation contains a further physical hypothesis,

the validity of which will hardly be doubted without

empirical e vddence to the contrary. It has been assumed

that all these clocksgo at the same vote if they are of

identical construction. Stated more exactly : When

two clocks arranged at rest in different places of a

reference-body are set in such a manner that a particular

position of the pointers of the one clock is simultaneous

(in the above sense) with the same position of the

pointers of the other clock, then identical "

settings"

are always simultaneous (in the sense of the above

definition).

IX

THE RELATIVITY OF SIMULTANEITY

UP to now our considerations have been referred

to a particular body of reference, which we

have styled a"

railway embankment." We

suppose a very long train travelling along the rails

with the constant velocity v and in the direction in

dicated in Fig. i. People travelling in this train will

with advantage use the train as a rigid reference-

body (co-ordinate system) ; they regard all events in

Train

reference to the train. Then every event which takes

place along the line also takes place at a particular

point of the train. Also the definition of simultaneity

can be given relative to the train in exactly the same

way as with respect to the embankment. As a natural

consequence, however, the following question arises :

Are two events (e.g.the two strokes of lightning A

and B) which are simultaneous with reference to the

railway embankment also simultaneous relativelyto the

train ? We shall show directly that the answer must

be in the negative.

When we say that the lightning strokes A and B are


simultaneous with respect to the embankment, we

mean : the rays of lightemitted at the placesA and

B, where the lightningoccurs, meet each other at the

mid-pointM of the lengthA " " B of the embankment.

But the events A and B also correspond to positionsA

and B on the train. Let M' be the mid-pointof the

distance A " " B on the travellingtrain. Just when

the flashes 1 of lightningoccur, this pointM' naturallycoincides with the pointM, but it moves towards the

rightin the diagram with the velocityv of the train. If

an observer sittingin the positionM' in the train did

not possess this velocity,then he would remain per

manently at M, and the light rays emitted by the

flashes of lightningA and B would reach him simul

taneously,i.e.they would meet justwhere he is situated.

Now in reality(consideredwith reference to the railway

embankment) he ishasteningtowards the beam of light

coming from B, whilst he is ridingon ahead of the beam

of lightcoming from A. Hence the observer will see

the beam of lightemitted from B earlier than he will

see that emitted from A. Observers who take the rail

way train as their reference-bodymust therefore come

to the conclusion that the lightningflash B took placeearlier than the lightningflash A. We thus arrive at

the important result :

Events which are simultaneous with reference to the

embankment are not simultaneous with respect to the

train,and vice versa (relativityof simultaneity).Every

reference-body(co-ordinatesystem)has itsown particulartime ; unless we are told the reference-bodyto which

the statement of time refers,there is no meaning in a

statement of the time of an event.

1 As judged from the embankment.

RELATIVITY OF SIMULTANEITY 27

Now before the advent of the theory of relativity

it had always tacitlybeen assumed in physics that the

statement of time had an absolute significance, i.e.

that it is independent of the state of motion of the body

of reference. But we have just seen that this assump

tion is incompatible with the most natural definition

of simultaneity ; if we discard this assumption, then

the conflict between the law of the propagation of

lightin vacua and the principle of relativity(developed

in Section VII) disappears.

We were led to that conflict by the considerations

of Section VI, which are now no longer tenable. In

that section we concluded that the man in the carriage,

who traverses the distance w per second relative to the

carriage,traverses the same distance also with respect to

the embankment in each second of time. But, according

to the foregoing considerations, the time required by a

particular occurrence with respect to the carriage must

not be considered equal to the duration of the same

occurrence as judged from the embankment (as refer

ence-body). Hence it cannot be contended that the

man in walking travels the distance w relative to the

railway line in a time which is equal to one second as

judged from the embankment.

Moreover, the considerations of Section VI are based

on yet a second assumption, which, in the light of a

strict consideration, appears to be arbitrary,although

it was always tacitly made even before the introduction

of the theory of relativity.

X

ON THE RELATIVITY OF THE CONCEPTION

OF DISTANCE

LET us consider two particular points on the train l

travelling along the embankment with the

velocity v, and inquire as to their distance apart.

We already know that it is necessary to have a body of

reference for the measurement of a distance, with respect

to which body the distance can be measured up. It is

the simplest plan to use the train itself as reference-

body (co-ordinate system). An observer in the train

measures, the interval by marking off his measuring-rod

in a straight line (e.g.along the floor of the carriage)

as many times as is necessary to take him from the one

marked point to the other. Then the number which

tells us how often the rod has to be laid down is the

required distance.

It is a different matter when the distance has to be

judged from the railway line. Here the following

method suggests itself. If we call A' and B' the two

points on the train whose distance apart is required,

then both of these points are moving with the velocity v

along the embankment. In the first place we require to

determine the points A and B of the embankment which

are just being passed by the two points A' and B' at a

1e.g. the middle of the first and of the hundredth carriage.

28

THE RELATIVITY OF DISTANCE 29

particular time t"

judged from the embankment.

These points A and B of the embankment can be deter

mined by applying the definition of time given in

Section VIII. The distance between these points A

and B is then measured by repeated application of the

measuring-rod along the embankment.

A priori it is by no means certain that this last

measurement will supply us with the same result as

the first. Thus the length of the train asmeasured

from the embankmentmay

be different from that

obtained by measuring in the train itself. This

circumstance leadsus to a second objection which must

be raised against the apparently obvious considera

tion of Section VI. Namely, if the man in the carriage

covers the distance w in a unit of time"

measured from

the train,"

then this distance" as measured from the

embankment"

is not necessarily also equal to w.

XI

THE LORENTZ TRANSFORMATION

THEresults of the last three sections show

that the apparent incompatibility of the law

of propagation of light with the principle of

relativity (Section VII) has been derived by means of

a consideration which borrowed two unjustifiable

hypotheses from classical mechanics; these are as

follows :

(1) The time-interval (time) between two events is

independent of the condition of motion of the

body of reference.

(2) The space-interval (distance) between two points

of a rigid body is independent of the condition

of motion of the body of reference.

If we drop these hypotheses, then the dilemma of

Section VII disappears, because the theorem of the addi

tion of velocities derived in Section VI becomes invalid.

The possibility presents itself that the law of the pro

pagation of light in vacuo may be compatible with the

principle of relativity, and the question arises : How

have we to modify the considerations of Section VI

in order to remove the apparent disagreement between

these two fundamental results of experience ? This

question leads to a general one. In the discussion of

THE LORENTZ TRANSFORMATION 31

Section VI we have to do with placesand times relative

both to the train and to the embankment. How are

we to find the placeand time of an event in relation to

the train, when we know the place and time of the

event with respect to the railway embankment ? Is

there a thinkable answer to this questionof such a

nature that the law of transmission of lightin vacua

does not contradict the principleof relativity? In

other words : Can we conceive of a relation between

placeand time of the individual events relative to both

reference-bodies,such that every ray of lightpossessesthe velocityof transmission c relative to the embank

ment and relative to the train ? This questionleads to

a quitedefinite positiveanswer, and to a perfectlydefinite

transformation law for the space-timemagnitudes of

an event when changingover from one body of reference

to another.

Before we deal with this, we shall introduce the

followingincidental consideration. Up to the present

we have only considered events takingplacealong the

embankment, which had mathematicallyto assume the

function of a straightline. In the manner indicated

in Section II we can imaginethis reference-bodysupplemented laterallyand in a vertical direction by means of

a framework of rods,so that an event which takes placeanywhere can be localised with reference to this frame

work. Similarly,we can imagine the train travellingwith the velocityv to be continued across the whole of

space, so that every event, no matter how far off it

may be,could also be localised with respectto the second

framework. Without committingany fundamental error,

we can disregardthe fact that in realitythese frame

works would continuallyinterfere with each other,owing


to the impenetrabilityof solid bodies. In every such

framework we imagine three surfaces perpendiculartoeach other marked out, and designatedas

" co-ordinate

planes "

(" co-ordinate system "). A co-ordinate

system K then corresponds to the embankment, and a

co-ordinate system K' to the train. An event, wherever

it may have taken place,would be fixed in space with

respect to K by the three perpendicularsx, y, z on the

co-ordinate planes,and with regard to time by a time-

x.value t. Relative to K', the

same event would be fixed

in respectof space and time

by corresponding values x't

y',z',t',which of course are

not identical with x, y, z,

t. It has already been set

forth in detail how these

magnitudes are to be re

garded as results of physicalmeasurements.

Obviously our problem can be exactly formulated in

the followingmanner. What are the values x,'y',z',t',

of an event with respect to K', when the magnitudes

x, y, z, t,of the same event with respect to K are given ?

The relations must be so chosen that the law of the

transmission of lightin vacua is satisfied for one and the

same ray of light (and of course for every ray) with

respect to K arid K'. For the relative orientation in

space of the co-ordinate systems indicated in the dia

gram (Fig.2),this problem is solved by means of the

equations:x-vt

FIG. 2.

A/ v*

VT --,


V*

c*

This system of equationsis known as the " Lorentz

transformation." l

If in placeof the law of transmission of lightwe had

taken as our basis the tacit assumptions of the older

mechanics as to the absolute character of times and

lengths,then instead of the above we should have

obtained the followingequations:

x'=x" vt

y'"yz'=z

t'=t.

This system of equationsis often termed the " Galilei

transformation." The Galilei transformation can be

obtained from the Lorentz transformation by sub

stitutingan infinitelylargevalue for the velocityof

lightc in the latter transformation.

Aided by the followingillustration,we can readilysee that, in accordance with the Lorentz transforma

tion, the law of the transmission of lightin vacua

is satisfied both for the reference-bodyK and for the

reference-bodyK'. A light-signalis sent along the

positive#-axis, and this light-stimulusadvances in

accordance with the equation

x=ct,

1 A simple derivation of the Lorentz transformation is givenin Appendix I.

3


i.e. with the velocity c. According to the equations of

the Lorentz transformation, this simple relation between

x and t involves a relation between x' and t'. In point

of fact, if we substitute for x the value ct in the first

and fourth equations of the Lorentz transformation,

we obtain :

_

/ V2

Vi-a

/V

V

I9cz

from which, by division, the expression

x'=ct'

immediately follows. If referred to the system K' ',the

propagation of light takes place according to this

equation. We thus see that the velocity of transmission

relative to the reference-body K' is also equal to c. The

same result is obtained for rays of light advancing in

any other direction whatsoever. Of course this is not

surprising, since the equations of the Lorentz trans

formation were derived conformably to this point of

view.

XII

THE BEHAVIOUR OF MEASURING-RODS AND

CLOCKS IN MOTION

IPLACE a metre-rod in the A;'-axis of K' in such a

manner that one end (the beginning) coincides with

the point x'=o, whilst the other end (the end of the

rod) coincides with the point x'=i. What is the length

of the metre-rod relativelyto the system K ? In order

to learn this, we need only ask where the beginning of the

rod and the end of the rod lie with respect to K at a

particulartime i of the system K. By means of the first

equation of the Lorentz transformation the values of

these two points at the time t=o can be shown to be

(beginning of rod)/T

"*V ]2

*(endof rod)

/ ifi\/ i -

-^the distance between the points being \/ i -

-^But

c

the metre-rod is moving with the velocity v relative to

K. It therefore follows that the length of a rigidmetre-

rod moving in the direction of its length with a velocity

v is "Ji" v2/cz of a metre. The rigid rod is thus

shorter when in motion than when at rest, and the

more quickly it is moving, the shorter is the rod. For

the velocity v"c we should have v/i-v2/c2=o, and

for still greater velocities the square-root becomes


imaginary. From this we conclude that in the theory

of relativitythe velocity c plays the part of a limiting

velocity, which can neither be reached nor exceeded

by any real body.

Of course this feature of the velocity c as a limiting

velocity also clearly follows from the equations of the

Lorentz transformation, for these become meaningless

if we choose values of v greater than c.

If, on the contrary, we had considered a metre-rod

at rest in the #-axis with respect to K, then we should

have found that the length of the rod as judged from

K' would have been J~L"VZ/CZ ; this is quite in accord

ance with the principleof relativitywhich forms the

basis of our considerations.

A prioriit is quite clear that we must be able to

learn something about the physicalbehaviour of measur

ing-rods and clocks from the equations of transforma

tion, for the magnitudes x, y, z, t, are nothing more nor

less than the results of measurements obtainable by

means of measuring-rods and clocks. If we had based

our considerations on the Galilei transformation we

should not have obtained a contraction of the rod as a

consequence of its motion.

Let us now consider a seconds-clock which is per

manently situated at the origin (x'=o) of K'. f=o

and t'=i are two successive ticks of this clock. The

first and fourth equations of the Lorentz transformation

give for these two ticks :

t = Q

and

RODS AND CLOCKS IN MOTION 87

As judged from K, the clock is moving with the

velocity v ; as judged from this reference-body, the

time which elapses between two strokes of the clock

i

is not one second, but / t"a seconds, i.e. a some-

what larger time. Asa consequence of its motion

the clock goes more slowly than when at rest. Here

also the velocity c plays the part of an unattainable

limiting velocity.

XIII

THEOREM OF THE ADDITION OF VELOCITIES.

THE EXPERIMENT OF FIZEAU

NOWin practice we can move clocks and

measuring-rods only with velocities that are

small compared with the velocity of light ; hence

we shall hardly be able to compare the results of the

previous section directly with the reality. But, on the

other hand, these results must strike you as being very

singular, and for that reason I shall now draw another

conclusion from the theory, one which can easily be

derived from the foregoing considerations, and which

has been most elegantly confirmed by experiment.

In Section VI we derived the theorem of the addition

of velocities in one direction in the form which also

results from the hypotheses of classical mechanics. This

theorem can also be deduced readily from the Galilei

transformation (Section XI). In place of the man

walking inside the carriage, we introduce a point moving

relatively to the co-ordinate system K' in accordance

with the equation

x'=wt'.

By means of the first and fourth equations of the Galilei

transformation we can express x' and t' in terms of x

and t, and we then obtain

x=(v-{-w)t.38

THE EXPERIMENT OF FIZEAU 89

This equationexpresses nothing else than the law of

motion of the point with reference to the system K

(ofthe man with reference to the embankment). We

denote this velocityby the symbol W, and we then

obtain, as in Section VI,

. . . (A).

But we can carry out this consideration just as well

on the basis of the theoryof relativity.In the equation

x'=wt'

we must then express x' and t'in terms of x and t,making

use of the first and fourth equations of the Lorentz

transformation.Instead of the equation (A) we then

obtain the equation

which corresponds to the theorem of addition for

velocities in one direction according to the theory of

relativity.The questionnow arises as to which of these

two theorems isthe better in accord with experience.On

this pointwe are enlightenedby a most importantexperiment which the brilliant physicistFizeau performedmore

than half a century ago, and which has been repeatedsince then by some of the best experimentalphysicists,so that there can be no doubt about its result. The

experiment is concerned with the followingquestion.

Light travels in a motionless liquidwith a particular

velocityw. How quicklydoes it travel in the direction

of the arrow in the tube T (seethe accompanying diagram,

Fig. 3) when the liquid above mentioned is flowing

through the tube with a velocityv ?


In accordance with the principleof relativitywe shall

certainlyhave to take for granted that the propagation

of lightalways takes place with the same velocityw

with respectto the liquid,whether the latter is in motion

with reference to other bodies or not. The velocityof lightrelative to the liquidand the velocityof the

latter relative to the tube are thus known, and we

requirethe velocityof lightrelative to the tube.

It is clear that we have the problem of Section VI

again before us. The tube playsthe part of the railwayembankment or of the co-ordinate system K, the liquid

plays the part of the carriageor of the co-ordinate

system K', and finally,the lightplays the part of the

FIG. 3.

man walking along the carriage,or of the moving point

in the presentsection. If we denote the velocityof the

light relative to the tube by W, then this is given

by the equation (A) or (B), according as the Galilei

transformation or the Lorentz transformation corre

sponds to the facts. Experiment l decides in favour

of equation(B)derived from the theoryof relativity,and

the agreement is,indeed, very exact. According to

1 Fizeau found W = w+v( i " 11,where n=- is the index of

refraction of the liquid. On the other hand, owing to the small-

ness of"5 as compared with i, we can replace (B) in the firstc

placeby W = (w +v)(i - -

Vor to the same order of approxima

tion by w +v( i "

2 j,which agrees with Fizeau's result.

THE EXPERIMENT OF FIZEAU 41

recent and most excellent measurements by Zeeman, the

influence of the velocity of flow v on the propagation of

light is represented by formula (B) to within one per

cent.

Nevertheless we must now draw attention to the fact

that a theory of this phenomenon was given by H. A.

Lorentz long before the statement of the theory of

relativity. This theory was of a purely electrody-

namical nature, and was obtained by the use of particular

hypotheses as to the electromagnetic structure of matter.

This circumstance, however, does not in the least

diminish the conclusiveness of the experiment as a

crucial test in favour of the theory of relativity, for the

electrodynamics of Maxwell-Lorentz, on which the

original theory was based, in no way opposes the theory

of relativity. Rather has the latter been developed

from electrodynamics as an astoundingly simple com

bination and generalisation of the hypotheses, formerly

independent of each other, on which electrodynamics

was built.

XIV

THE HEURISTIC VALUE OF THE THEORY OF

RELATIVITY

OURtrain of thought in the foregoing pages can be

epitomised in the following manner. Experience

has led to the conviction that, on the one hand,

the principle of relativity holds true, and that on the

other hand the velocity of transmission of light in vacuo

has to be considered equal to a constant c. By uniting

these two postulates we obtained the law of transforma

tion for the rectangular co-ordinates x, y, z and the time

t of the events which constitute the processes of nature.

In this connection we did not obtain the Galilei trans

formation, but, differing from classical mechanics,

the Lorentz transformation.

The law of transmission of light, the acceptance of

which is justified by our actual knowledge, played an

important part in this process of thought. Once in

possession of the Lorentz transformation, however,

we can combine this with the principle of relativity,

and sum up the theory thus :

Every general law of nature must be so constituted

that it is transformed into a law of exactly the same

form when, instead of the space-time variables x, y, z, t

of the original co-ordinate system K, we introduce new

space-time variables x', y', z', t' of a co-ordinate system

HEURISTIC VALUE OF RELATIVITY 43

.

In this connection the relation between the

"rdinary and the accented magnitudes is given by the

Lorentz transformation. Or, in brief : General laws

"f nature are co-variant with respect to Lorentz trans-

ormations.

This is adefinite mathematical condition that the

heory of relativity demands of a natural law, and in

virtue of this, the theory becomesa valuable heuristic aid

n the search for general laws of nature. Ifa general

awof nature were to be found which did not satisfy

his condition, then at leastone

of the two fundamental

assumptions of the theory would have been disproved,

-et us now examine what general results the latter

heory has hitherto evinced.

XV

GENERAL RESULTS OF THE THEORY

ITis clear from our previous considerations that the

(special)theory of relativityhas grown out of electro

dynamics and optics. In these fields it has not

appreciably altered the predictions of theory, but it

has considerably simplified the theoretical structure,

i.e. the derivation of laws, and"

what is incomparably

more important "it has considerably reduced the

number of independent hypotheses forming the basis of

theory. The special theory of relativity has rendered

the Maxwell-Lorentz theory so plausible, that the latter

would have been generally accepted by physicists

even if experiment had decided less unequivocally in its

favour.

Classical mechanics required to be modified before it

could come into line with the demands of the special

theory of relativity. For the main part, however,

this modification affects only the laws for rapid motions,

in which the velocities of matter v are not very small as

compared with the velocity of light. We have experi

ence of such rapid motions only in the case of electrons

and ions ; for other motions the variations from the laws

of classical mechanics are too small to make themselves

evident in practice. We shall not consider the motion

of stars until we come to speak of the general theory of

relativity. In accordance with the theory of relativity

GENERAL RESULTS OF THEORY 45

ic kinetic energy of a material point of mass m is no

mger given by the well-known expression

m-.

ut by the expressionme2

"Ta7~~v*V1-!

e"

'his expressionapproaches infinityas the velocityv

pproachesthe velocityof lightc. The velocitymust

herefore always remain less than c, however great may

e the energiesused to produce the acceleration. If

/e develop the expressionfor the kinetic energy in the

arm of a series,we obtain

5-"-- ....

8 c2

v2When

-2is small compared with unity, the third

0

"i these terms is always small in comparison with the

econd, which last is alone considered in classical

aechanics. The first term me2 does not contain

he velocity,and requiresno consideration if we are only

leaiingwith the questionas to how the energy of a

"oint-mass depends on the velocity. We shall speak)f its essential significancelater.

The most important result of a generalcharacter to

vhich the specialtheory of relativityhas led is concerned

vith the conceptionof mass. Before the advent of

"elativity,physicsrecognisedtwo conservation laws of

'undamental importance, namely, the law of the con

servation of energy and the law of the conservation of

nass ; these two fundamental laws appearedto be quite


independent of each other. By means of the theoryofjrelativitythey have been united into one law. We shall

now brieflyconsider how this unification came about,

and what meaning is to be attached to it.

The principleof relativityrequiresthat the law of the jconservation of energy should hold not only with re

ference to a co-ordinate system K, but also with respect

to every co-ordinate system K' which is in a state of

uniform motion of translation relative to K, or, briefly,relative to every

" Galileian "

system of co-ordinates.

In contrast to classical mechanics, the Lorentz trans

formation is the decidingfactor in the transition from

one such system to another.

By means of comparatively simple considerations

we are led to draw the followingconclusion from

these premises,in conjunctionwith the fundamental

equationsof the electrodynamicsof Maxwell : A bodymoving with the velocityv, which absorbs 1

an amount

of energy E0 in the form of radiation without sufferingan alteration in velocityin the process, has, as a conse

quence, its energy increased by an amount

!?

In consideration of the expressiongivenabove for the

kinetic energy of the body, the requiredenergy of the

body comes out to be

1 E0 is the energy taken up, as judged from a co-ordinate

system moving with the body.

GENERAL RESULTS OF THEORY 47

Thus the body has the same energy as a body of mass

-fj)moving with the velocityv. Hence we can

say : If a body takes up an amount of energy ZT0,then

"its inertial mass increases by an amount ~| ; the

inertial mass of a body is not a constant, but varies

according to the change in the energy of the body.

The inertial mass of a system of bodies can even be

regarded as a measure of its energy. The law of the

conservation of the mass of a system becomes identical

with the law of the conservation of energy, and is onlyvalid providedthat the system neither takes up nor sends

out energy. Writing the expressionfor the energy in

the form

I " ~Jjjc2

we see that the term we2, which has hitherto attracted

our attention,is nothing else than the energy possessed

by the body * before it absorbed the energy E0.A direct comparison of this relation with experiment

is not possibleat the presenttime, owing to the fact that

the changes in energy E0 to which we can subjecta

system are not large enough to make themselves

perceptibleas a change in the inertial mass of the

system. -f is too small in comparison with the massC

m, which was presentbefore the alteration of the energy.

It is owing to this circumstance that classical mechanics

was able to establish successfullythe conservation of

mass as a law of independentvalidity.1 As judged from a co-ordinate system moving with the body.


Let me add a final remark of a fundamental nature.

The success of the Faraday-Maxwell interpretation of

electromagnetic action at a distance resulted in physicists

becoming convinced that there are no such things as

instantaneous actions at a distance (not involving an

intermediary medium) of the type of Newton's law of

gravitation. According to the theory of relativity,

action at a distance with the velocity of light always

takes the place of instantaneous action at a distance or

of action at a distance with an infinite velocity of trans

mission. This is connected with the fact that the

velocity c plays a fundamental role in this theory. In

Part II we shall see in whatway

this result becomes

modified in the general theory of relativity.

XVI

EXPERIENCE AND THE SPECIAL THEORY OF

RELATIVITY


latter effect manifests itself in a slightdisplacement

of the spectrallines of the lighttransmitted to us from

a fixed star, as compared with the positionof the same

spectrallines when they are produced by a terrestrial

source of light(Doppler principle).The experimental

arguments in favour of the Maxwell-Lorentz theory,which are at the same time arguments in favour of the

theory of relativity,are too numerous to be set forth

here. In realitythey limit the theoretical possibilitiesto such an extent, that no other theory than that of

Maxwell and Lorentz has been able to hold its own when

tested by experience.But there are two classes of experimental facts

hitherto obtained which can be representedin the

Maxwell-Lorentz theory only by the introduction of an

auxiliaryhypothesis,which in itself" i.e. without

making use of the theory of relativity" appears ex

traneous.

It is known that cathode rays and the so-called

/2-raysemitted by radioactive substances consist of

negativelyelectrified particles(electrons)of very small

inertia and largevelocity. By examining the deflection

of these rays under the influence of electric and magnetic

fields,we can study the law of motion of these particles

very exactly.In the theoretical treatment of these electrons,we are

faced with the difficultythat electrodynamictheory of

itselfis unable to give an account of their nature. For

since electrical masses of one signrepeleach other, the

negativeelectrical masses constitutingthe electron would

necessarilybe scattered under the influence of their

mutual repulsions,unless there are forces of another

kind operatingbetween them, the nature of which has

EXPERIENCE AND RELATIVITY 51

litherto remained obscure to us.1 If we now assume

:hat the relative distances between the electrical masses

Constitutingthe electron remain unchanged during the

motion of the electron (rigidconnection in the sense of

classical mechanics),we arrive at a law of motion of the

electron which does not agree with experience. Guided

oy purelyformal pointsof view, H. A. Lorentz was the

first to introduce the hypothesis that the particles

:onstitutingthe electron experience a contraction

in the direction of motion in consequence of that motion,

ithe amount of this contraction being proportionalto

/ v2the expression\f i "

%.This hypothesis,which is

" c

not justifiableby any electrodynamicalfacts,suppliesus'then with that particularlaw of motion which has

been confirmed with great precisionin recent years.

The theory of relativityleads to the same law of

motion, without requiringany specialhypothesiswhat

soever as to the structure and the behaviour of the

electron. We arrived at a similar conclusion in Section

XIII in connection with the experiment of Fizeau, the

result of which is foretold by the theory of relativitywithout the necessityof drawing on hypotheses as to

the physicalnature of the liquid.The second class of facts to which we have alluded

has reference to the questionwhether or not the motion

of the earth in space can be made perceptiblein terrestrial

experiments. We have alreadyremarked in Section V

that all attempts of this nature led to a negativeresult.

Before the theory of relativitywas put forward, it was

1 The general theory of relativityrenders it likelythat the

electrical masses of an electron are held together by gravitational forces.


difficult to become reconciled to this negative result,

for reasons now to be discussed. The inherited

prejudicesabout time and space did not allow any

doubt to arise as to the prime importance of the

Galilei transformation for changing over from one

body of reference to another. Now assuming that the

Maxwell-Lorentz equationshold for a reference-bodyK,

we then find that they do not hold for a reference-

body K' moving uniformly with respect to K, if we

assume that the relations of the Galileian transforma

tion exist between the co-ordinates of K and K'. It

thus appears that of all Galileian co-ordinate systems

one (K) correspondingto a particularstate of motion

is physicallyunique. This result was interpreted

physicallyby regardingK as at rest with respect to a

hypotheticalaether of space. On the other hand,

all co-ordinate systems K' moving relativelyto K were

to be regarded as in motion with respect to the aether.

To this motion of K' againstthe aether ("aether-drift"

relative to K'} were assignedthe more complicatedlaws which were supposed to hold relative to K'.

Strictlyspeaking,such an aether-drift ought also to be

assumed relative to the earth, and for a long time the

efforts of physicistswere devoted to attempts to detect

the existence of an aether-drift at the earth's surface.

In one of the most notable of these attempts Michelson

devised a method which appears as though it must be

decisive. Imagine two mirrors so arranged on a rigidbody that the reflectingsurfaces face each other. A

ray of lightrequiresa perfectlydefinite time T to pass

from one mirror to the other and back again,ifthe whole

system be at rest with respectto the aether. It is found

by calculation,however, that a slightlydifferent time

EXPERIENCE AND RELATIVITY 53

is requiredfor this process, if the body,togetherwith

:he mirrors, be moving relativelyto the aether. And

another point: it is shown by calculation that for

i given velocityv with reference to the aether,this

ime T' is different when the body is moving perpen-

licularlyto the planesof the mirrors from that resultingvhen the motion is parallelto these planes. Althoughhe estimated difference between these two times is

exceedinglysmall, Michelson and Morley performed an

experimentinvolvinginterference in which this difference

should have been clearlydetectable. But the experiment gave a negativeresult " a fact very perplexing:o physicists.Lorentz and FitzGerald rescued the

:heoryfrom this difficultyby assuming that the motion

)f the body relative to the aether producesa contraction

Df the body in the direction of motion, the amount of con

traction beingjustsufficient to compensate for the differ-

snce in time mentioned above. Comparison with the

liscussion in Section XII shows that also from the stand

point of the theory of relativitythis solution of the

difficultywas the right one. But on the basis of the

theory of relativitythe method of interpretationis

incomparably more satisfactory.According to this

theorythere is no such thingas a"

speciallyfavoured"

(unique)co-ordinate system to occasion the introduction

of the aether-idea,and hence there can be no aether-drift,

nor any experiment with which to demonstrate it.

Here the contraction of moving bodies follows from

the two fundamental principlesof the theory without

the introduction of particularhypotheses; and as the

prime factor involved in this contraction we find,not

the motion in itself,to which we cannot attach any

meaning, but the motion with respect to the body of


reference chosen in the particular case in point. Thus

for a co-ordinate system moving with the earth the

mirror system of Michelson and Morley is not shortened,

but it is shortened for aco-ordinate S}'stein which is at

rest relatively to the sun.

XVII

MINKOWSKFS FOUR-DIMENSIONAL SPACE

THE non-mathematician is seized by a mysterious

shuddering when he hears of " four-dimensional"

tilings,by a feeling not unlike that awakened by

thoughts of the occult. And yet there is no more

common-place statement than that the world in which

we live is a four-dimensional space-time continuum.

Space is a three-dimensional continuum. By this

we mean that it is possible to describe the position of a

point (at rest) by means of three numbers (co-ordinates)

x, y, z, and that there is an indefinite number of points

in the neighbourhood of this one, the position of which

can be described by co-ordinates such as xlf ylt zlt which

may be as near as we choose to the respective values of

the co-ordinates x, y, z of the first point. In virtue of the

latter property we speak of a" continuum," and owing

to the fact that there are three co-ordinates we speak of

it as being " three-dimensional."

Similarly, the world of physical phenomena which was

briefly called " world"

by Minkowski is naturally

four-dimensional in the space-time sense. For it is

composed of individual events, each of which is de

scribed by four numbers, namely, three space

co-ordinates x, y, z and a time co-ordinate, the time-

value t. The " world " is in this sense also a continuum ;

for to every event there are as many" neighbouring

"

55


events (realisedor at least thinkable)as we care to

choose, the co-ordinates xlt ylt zlt ^ of which differ

by an indefinitelysmall amount from those of the

event x, y, z, t originallyconsidered. That we have not

been accustomed to regard the world in this sense as a

four-dimensional continuum is due to the fact that in

physics,before the advent of the theory of relativity,time played a different and more independentrole,as

compared with the space co-ordinates. It is for this

reason that we have been in the habit of treatingtime

as an independent continuum. As a matter of fact,

according to classical mechanics, time is absolute,

i.e.it is independentof the positionand the condition

of motion of the system of co-ordinates. We see this

expressed in the last equation of the Galileian trans

formation (t'=t}.The four-dimensional mode of consideration of the

"

world " is natural on the theory of relativity,since

accordingto this theorytime is robbed of its independence. This is shown by the fourth equation of the

Lorentz transformation :

Moreover, accordingto this equationthe time difference

A"' of two events with respectto K' does not in general

vanish, even when the time difference A2 of the same

events with reference to K vanishes. Pure "

space-

distance " of two events with respect to K results in" time-distance "

of the same events with respectto K'.

But the discoveryof Minkowski, which was of import-

FOUR-DIMENSIONAL SPACE 57

ance for the formal development of the theory of re

lativity,does not lie here. It is to be found rather in

the fact of his recognitionthat the four-dimensional

space-timecontinuum of the theoryof relativity,in its

most essential formal properties,shows a pronounced

relationshipto the three-dimensional continuum of

Euclidean geometricalspace.1 In order to givedue

prominence to this relationship,however, we must

replacethe usual time co-ordinate t by an imaginary

magnitude \l-i.ct proportionalto it. Under these

conditions,the natural laws satisfyingthe demands of

the (special)theory of relativityassume mathematical

forms, in which the time co-ordinate playsexactlythe

same role as the three space co-ordinates. Formally,these four co-ordinates correspondexactlyto the three

space co-ordinates in Euclidean geometry. It must be

clear even to the non-mathematician that, as a conse

quence of this purelyformal addition to our knowledge,the theory perforcegained clearness in no mean

measure.

These inadequateremarks can givethe reader only a

vague notion of the importantidea contributed by Min-

kowski. Without it the generaltheoryof relativity,ofwhich the fundamental ideas are developedin the follow

ing pages, would perhaps have got no farther than its

longclothes. Minkowski's work is doubtless difficultof

access to anyone inexperiencedin mathematics, but

since it is not necessary to have a very exact grasp of

this work in order to understand the fundamental ideas

of either the specialor the generaltheoryof relativity,I shall at present leave it here, and shall revert to it

onlytowards the end of Part II.

1 Cf.the somewhat more detailed discussion in Appendix II.

PART II


XVIII

SPECIAL AND GENERAL PRINCIPLE OF

RELATIVITY

THE basal principle, which was the pivot of all

our previous considerations, was the special

principle of relativity, i.e. the principle of the

physical relativity of all uniform motion. Let us once

more analyse its meaning carefully.

It was at all times clear that, from the point of view

of the idea it conveys to us, every motion must only

be considered as a relative motion. Returning to the

illustration we have frequently used of the embankment

and the railway carriage, we can express the fact of the

motion here taking place in the following two forms,

both of which are equally justifiable:

(a) The carriage is in motion relative to the embank

ment.

(b) The embankment is in motion relative to the

carriage.

In (a) the embankment, in (b) the carriage, serves as

the body of reference in our statement of the motion

taking place. If it is simply a question of detecting59

60 GENERAL THEORY OF RELATIVITY

or of describingthe motion involved, it is in principleimmaterial to what reference-bodywe refer the motion.

As alreadymentioned, this is self-evident,but it must

not be confused with the much more comprehensive

statement called " the principleof relativity,"which

we have taken as the basis of our investigations.The principlewe have made use of not onlymaintains

that we may equallywell choose the carriageor the

embankment as our reference-bodyfor the descriptionof any event (forthis,too, isself-evident).

Our principlerather asserts what follows : If we formulate the generallaws of nature as they are obtained from experience,

by making use of

(a)the embankment as reference-body,

(b)the railwaycarriageas reference-body,then these generallaws of nature (e.g.the laws of

mechanics or the law of the propagationof lightin vacua)have exactly the same form in both cases. This can

also be expressedas follows : For the physicaldescription of natural processes, neither of the reference-

bodies K, K' is unique (lit."

speciallymarked out ") as

compared with the other. Unlike the first,this latter

statement need not of necessityhold a priori; it is

not contained in the conceptions of " motion " and"

reference-body" and derivable from them ; only

experiencecan decide as to its correctness or incor

rectness.

Up to the present,however, we have by no means

maintained the equivalenceof all bodies of reference K

in connection with the formulation of natural laws.

Our course was more on the followinglines. In the

first place,we started out from the assumption that

there exists a reference-bodyK, whose condition of

SPECIAL AND GENERAL PRINCIPLE 61

motion is such that the Galileian law holds with respectto it : A particleleftto itselfand sufficientlyfar removed

from all other particlesmoves uniformly in a straightline. With reference to K (Galileianreference-body)the

laws of nature were to be as simple as possible. But

in addition to K, all bodies of reference K' should be

givenpreferencein this sense, and theyshould be exactly

equivalentto K for the formulation of natural laws,

provided that they are in a state of uniform rectilinear

and non-rotary motion with respect to K ; all these

bodies of reference are to be regarded as Galileian

reference-bodies. The validityof the principleof

relativitywas assumed only for these reference-bodies,

but not for others (e.g.those possessingmotion of a

different kind). In this sense we speak of the special

principleof relativity,or specialtheoryof relativity.In contrast to this we wish to understand by the

"

generalprincipleof relativity" the followingstate

ment : All bodies of reference K, K', etc., are equivalentfor the descriptionof natural phenomena (formulationof

the general laws of nature),whatever may be their

state of motion. But before proceeding farther, it

ought to be pointed out that this formulation must be

replacedlater by a more abstract one, for reasons which

will become evident at a later stage.Since the introduction of the specialprincipleof

relativityhas been justified,every intellect which

strives after generalisationmust feel the temptationto venture the step towards the general principleof

relativity.But a simple and apparently quite reliable

consideration seems to suggest that, for the presentat any rate, there is little hope of success in such an

attempt. Let us imagine ourselves transferred to our


old friend the railway carriage, which is travelling at a

uniform rate. As long as it is moving uniformly, the

occupant of the carriage is not sensible of its motion,

and it is for this reason that he can without reluctance

interpret the facts of the case as indicating that the

carriage is at restobut the embankment in motion.

Moreover, according to the special principle of relativity,

this interpretation is quite justifiedalso from a physical

point of view.

If the nlotion of the carriage is now changed into a

non-uniform; motion, as for instance by a powerful

application of the brakes, then the occupant of the

carriage experiences a correspondingly powerful jerk

forwards. The retarded motion is manifested in the

mechanical behaviour of bodies relative to the person

in the railway carriage. The mechanical behaviour is

different from that of the case previously considered,

and for this reason it would appear to be impossible

that the same mechanical laws hold relativelyto the non-

uniformly moving carriage,as hold with reference to the

carriage when at rest or in uniform motion. At all

events it is clear that the Galileian law does not hold

with respect to the non-uniformly moving carriage.

Because of this, we feel compelled at the present juncture

to grant a kind of absolute physical reality to non-

uniform motion, in opposmpn to the general principle

of relativity. But in what follows we shall soon see

that this conclusion cannot be maintained.

XIX

THE GRAVITATIONAL FIELD

IFwe pick up a stone and then let it go, why does it

fall to the ground ?" The usual answer to this

question is^f *" Because it is attracted by the earth."

Modern physics formulates the answer rather differently

for the following reason. As a result of the more careful

study of electromagnetic phenomena, we have come

to regard action at a distance as a process impossible

without the intervention of some intermediary medium.

If, for instance, a mag^lt attracts a piece of iron, we

cannot be content to regard this as meaning that the

magnet acts directly op the iron through the inter

mediate empty space, tout we are constrained to im

agine "after the manner of Faraday "

that the magnet

always calls into being something physically real in

the space around it, that something being what we call a

"

magnetic field." In its turn this magnetic field

operates on the piece of iron, so that the latter strives

to move towards the magnet. We shall not discuss

here the justification for this incidental conception,

which is indeed a somewhat arbitrary one. We shall

only mention that with its aid electromagnetic pheno

mena can be theoretically represented much more

satisfactorily than without it, and this applies partic

ularly to the transmission of electromagnetic waves.

63


The effects of gravitationalso are regardedin an ana

logous manner.

The action of the earth on the stone takes placein

directly.The earth produces in its surroundings a

gravitationalfield,which acts on the stone and producesits motion of fall. As we know from experience,the

intensityof the action on a body diminishes accordingto a quitedefinite law, as we proceedfarther and farther

away from the earth. From our point of view this

means : The law governingthe propertiesof the gravitational field in space must be a perfectlydefinite one, in

order correctlyto represent the diminution of gravitational action with the distance from operativebodies.

It is something like this : The body (e.g.the earth)produces a field in its immediate neighbourhood directly;the intensityand direction of the field at pointsfarther

removed from the body are thence determined bythe law which governs the propertiesin space of the

gravitationalfields themselves.

In contrast to electric and magneticfields,the gravitational field exhibits a most remarkable property, which

is of fundamental importance for what follows. Bodies

which are moving under the sole influence of a gravitational field receive an acceleration,which does not in the

least depend either on the material or on the physicalstate of the body. For instance, a piece of lead and

a piece of wood fall in exactlythe same manner in a

gravitationalfield (invacuo),when they start off from

rest or with the same initial velocity. This law, which

holds most accurately,can be expressedin a different ^

form in the lightof the followingconsideration.

Accordingto Newton's law of motion, we have

(Force)=(inertialmass)X (acceleration),

65

where the " inertial mass" is a characteristic constant

of the accelerated body. If now gravitation is the

cause of the acceleration, we then have

(Force)= (gravitationalmass) X (intensityof the

gravitationalfield),

where the "

gravitational mass"

is likewise a character

istic constant for the body. From these two relations

follows :

(gravitational mass)(acceleration)="

^^ mass)"X (intensityof the

gravitationalfield).

If now, as we find from experience,the acceleration is

to be independent of the nature and the condition of the

body and always the same for a given gravitational

field,then the ratio of the gravitationalto the inertial

mass must likewise be the same for all bodies. By a

suitable choice of units we can thus make this ratio

equal to unity. We then have the following law :

The gravitationalmass of a body is equal to its inertial

mass.

It is true that this important law had hitherto been

recorded in mechanics, but it had not been interpreted.

A satisfactoryinterpretationcan be obtained only if we

recognise the following fact : The same quality of a

body manifests itself according to circumstances as

" inertia" or as"

weight" (lit.

" heaviness "). In the

following section we shall show to what extent this is

actually the case, and how this question is connected

with the generalpostulateof relativity.

XX

THE EQUALITY OF INERTIAL AND GRAVITA

TIONAL MASS AS AN ARGUMENT FOR THE

GENERAL POSTULATE OF RELATIVITY

WE imagine a large portion of empty space, so far

removed from stars and other appreciable

masses, that we have before us approximately

the conditions required by the fundamental law of Galilei.

It is then possible to choose a Galileian reference-body for

this part of space (world), relative to which points at

rest remain at rest and points in motion continue per

manently in uniform rectilinear motion. As reference-

body let us imagine a spacious chest resembling a room

with an observer inside who is equipped with apparatus.

Gravitation naturally does not exist for this observer.

He must fasten himself with strings to the floor,

otherwise the slightest impact against the floor will

cause him to rise slowly towards the ceiling of the

room.

To the middle of the lid of the chest is fixed externally

a hook with rope attached, and now a"

being"

(what

kind of a being is immaterial to us) begins pulling at

this with a constant force. The chest together with the

observer then begin to move"

upwards" with a

uniformly accelerated motion. In course of time their

velocity will reach unheard-of values" provided that

66

INERTIAL AND GRAVITATIONAL MASS 67

we are viewing all this from another reference-bodywhich is not being pulledwith a rope.

But how does the man in the chest regardthe process ?

The acceleration of the chest will be transmitted to him

by the reaction of the floor of the chest. He must

therefore take up this pressure by means of his legsif

he does not wish to be laid out full length on the floor.

He is then standingin the chest in exactlythe same way

as anyone stands in a room of a house on our earth.

If he release a body which he previouslyhad in his

hand, the acceleration of the chest will no longerbe

transmitted to this body, and for this reason the bodywill approach the floor of the chest with an accelerated

relative motion. The observer will further convince

himself that the acceleration of the body towards the floor

of the chest is always of the same magnitude,whatever

kind ofbodyhe may happen to use forthe experiment.

Relying on his knowledge of the gravitationalfield

(asit was discussed in the precedingsection),the man

in the chest will thus come to the conclusion that he

and the chest are in a gravitationalfieldwhich isconstant

with regard to time. Of course he will be puzzled for

a moment as to why the chest does not fall,in this

gravitationalfield. Just then, however, he discovers

the hook in the middle of the lid of the chest and the

rope which is attached to it,and he consequentlycomes

to the conclusion that the chest is suspended at rest in

the gravitationalfield.

Ought we to smile at the man and say that he errs

in his conclusion ? I do not believe we ought to if we

wish to remain consistent ; we must rather admit that

his mode of graspingthe situation violates neither reason

nor known mechanical laws. Even though it is being


accelerated with respect to the " Galileian space"

first considered, we can nevertheless regard the chest

as being at rest. We have thus good grounds for

extending the principleof relativityto include bodies

of reference which are accelerated with respect to each

other,and as a result we have gaineda powerfulargumentfor a generalisedpostulateof relativity.

We must note carefullythat the possibilityof this

mode of interpretationrests on the fundamental

property of the gravitationalfield of giving all bodies

the same acceleration,or, what comes to the same thing,

on the law of the equalityof inertial and gravitational

mass. If this natural law did not exist, the man in

the accelerated chest would not be able to interpretthe behaviour of the bodies around him on the supposition of a gravitationalfield,and he would not be justified

on the grounds of experiencein supposinghis reference-

body to be "

at rest."

Suppose that the man in the chest fixes a rope to the

inner side of the lid,and that he attaches a body to the

free end of the rope. The result of this will be to stretch

the rope so that it will hang"

vertically"

downwards.

If we ask for an opinion of the cause of tension in the

rope, the man in the chest will say :" The suspended

body experiencesa downward force in the gravitational

field,and this is neutralised by the tension of the rope ;

what determines the magnitude of the tension of the

rope is the gravitationalmass of the suspended body."On the other hand, an observer who is poisedfreelyin

space will interpretthe condition of thingsthus :" The

rope must perforcetake part in the accelerated motion

of the chest, and it transmits this motion to the bodyattached to it. The tension of the rope is just large

INERTIAL AND GRAVITATIONAL MASS 69

enough to effect the acceleration of the body. That

which determines the magnitude of the tension of the

rope is the inertial mass of the body." Guided bythis example, we see that our extension of the principleof relativityimplies the necessityof the law of the

equalityof inertial and gravitationalmass. Thus we

have obtained a physicalinterpretationof this law.

From our consideration of the accelerated chest we

see that a generaltheory of relativitymust yieldim

portant results on the laws of gravitation. In point of

fact, the systematicpursuit of the general idea of re

lativityhas suppliedthe laws satisfied by the gravitational field. Before proceeding farther, however, I

must warn the reader againsta misconceptionsuggested

by these considerations. A gravitationalfield exists

for the man in the chest, despitethe fact that there was

no such field for the co-ordinate system first chosen.

Now we might easilysuppose that the existence of a

gravitationalfield is always only an apparent one. We

might also think that, regardlessof the kind of gravita

tional field which may be present, we could alwayschoose another reference-bodysuch that no gravitationalfield exists with reference to it. This is by no means

true for all gravitationalfields,but only for those of

quite specialform. It is, for instance, impossibleto

choose a body of reference such that, as judged from it,

the gravitationalfield of the earth (in its entirety)vanishes.

We can now appreciatewhy that argument is not

convincing,which we brought forward against the

generalprincipleof relativityat the end of Section XVIII.

It is certainlytrue that the observer in the railway

carriageexperiencesa jerk forwards as a result of the


application of the brake, and that he recognises in this the

non-uniformity of motion (retardation) of the carriage.

But he is compelled by nobody to refer this jerk to a

" real" acceleration (retardation) of the carriage. He

might also interpret his experience thus :"

My body of

reference (the carriage) remains permanently at rest.

With reference to it, however, there exists (during the

period of application of the brakes) a gravitational

field which is directed forwards and which is variable

with respect to time. Under the influence of this field,

the embankment together with the earth moves non-

uniformly in such a manner that their original velocity

in the backwards direction is continuously reduced."

XXI

IN WHAT RESPECTS ARE THE FOUNDATIONS

OF CLASSICAL MECHANICS AND OF THE

SPECIAL THEORY OF RELATIVITY UN

SATISFACTORY ?

WE have already stated several times that

classical mechanics starts out from the follow

ing law : Material particles sufficiently far

removed from other material particles continue to

move uniformly in a straight line or continue in a

state of rest. We have also repeatedly emphasised

that this fundamental law can only be valid for

bodies of reference K which possess certain unique

states of motion, and which are in uniform translational

motion relative to each other. Relative to other refer

ence-bodies K the law is not valid. Both in classical

mechanics and in the special theory of relativity we

therefore differentiate between reference-bodies K

relative to which the recognised" laws of nature

"

can

be said to hold, and reference-bodies K relative to which

these laws do not hold.

But no person whose mode of thought is logical can

rest satisfied with this condition of things. He asks :

" How does it come that certain reference-bodies (or

their states of motion) are given priority over other

reference-bodies (or their states of motion) ? What is


the reason for this preference? In order to show clearly

what I mean by this question,I shall make use of a

comparison.I am standing in front of a gas range. Standing

alongsideof each other on the range are two pans so

much alike that one may be mistaken for the other.

Both are half full of water. I notice that steam isbeing

emitted continuouslyfrom the one pan, but not from the

other. I am surprisedat this,even if I have never seen

either a gas range or a pan before. But if I now notice

a luminous something of bluish colour under the first

pan but not under the other, I cease to be astonished,

even if I have never before seen a gas flame. For I

can only say that this bluish something will cause the

emission of the steam, or at least possiblyit may do so.

If, however, I notice the bluish something in neither

case, and if I observe that the one continuouslyemits

steam whilst the other does not, then I shall remain

astonished and dissatisfied until I have discovered

some circumstance to which I can attribute the different

behaviour of the two pans.

Analogously,I seek in vain for a real something in

classical mechanics (or in the specialtheory of rela

tivity)to which I can attribute the different behaviour

of bodies considered with respect to the reference-

systems K and K'.1 Ne,\vton saw this objectionand

attempted to invalidate it,but without success. But

E. Mach recognisedit most clearlyof all,and because

of this objectionhe claimed that mechanics must be

1 The objection is of importance more especiallywhen the state

of motion of the reference-bodyis of such a nature that it does

not require any external agency for its maintenance, e.g. in

the case when the reference-bodyis rotatinguniformly.

MECHANICS AND RELATIVITY 73

placed on a new basis. It can only be got rid of by

means of a physics which is conformable to the general

principle of relativity, since the equations of such a

theory hold for every body of reference, whatever

maybe its state of motion.

XXII

A FEW INFERENCES FROM THE GENERAL

PRINCIPLE OF RELATIVITY

THE considerations of Section XX show that the

general principle of relativityputs us in a position

to derive properties of the gravitational field in a

purely theoretical manner. Let us suppose, for instance,

that we know the space-time "

course" for any natural

process whatsoever, as regards the manner in which it

takes place in the Galileian domain relative to a

Galileian body of reference K. By means of purely

theoretical operations (i.e.simply by calculation) we are

then able to find how this known natural process

appears, as seen from a reference-body K' which is

accelerated relatively to K. But since a gravitational

field exists with respect to this new body of reference

K', our consideration also teaches us how the gravita

tional field influences the process studied.

For example, we learn that a body which is in a state

of uniform rectilinear motion with respect to K (in

accordance with the law of Galilei) is executing an

accelerated and in general curvilinear motion with

respect to the accelerated reference-body K' (chest).

This acceleration or curvature corresponds to the in

fluence on the moving body of the gravitational field

prevailing relatively to K'. It is known that a gravita

tional field influences the movement of bodies in this

INFERENCES FROM RELATIVITY 75

way, so that our consideration suppliesus with nothing

essentiallynew.

However, we obtain a new result of fundamental

importance when we carry out the analogous considera

tion for a ray of light. With respect to the Galileian

reference-bodyK, such a ray of lightis transmitted

rectilinearlywith the velocityc. It can easilybe shown

that the path of the same ray of lightis no longer a

straightline when we consider it with reference to the

accelerated chest (reference-bodyK'). From this we

conclude, that, in general,rays of lightare propagated

curvilinearlyin gravitationalfields.In two respectsthis result is of great importance.

In the firstplace,it can be compared with the reality.

Although a detailed examination of the questionshows

that the curvature of lightrays requiredby the general

theory of relativityis only exceedinglysmall for the

gravitationalfields at our disposalin practice,its esti

mated magnitude for light rays passing the sun at

grazing incidence is nevertheless 1*7 seconds of arc.

This ought to manifest itself in the followingway.As seen from the earth, certain fixed stars appear to be

in the neighbourhoodof the sun, and are thus capableof observation during a total eclipseof the sun. At such

times, these stars ought to appear to be displacedoutwards from the sun by an amount indicated above,

as compared with their apparent positionin the skywhen the sun is situated at another part of the heavens.

The examination of the correctness or otherwise of this

deduction is a problem of the greatest importance,the

earlysolution of which is to be expected of astronomers.1

1 By means of the star photographs of two expeditionsequippedby a Joint Committee of the Royal and Royal Astronomical


In the second place our result shows that,accordingto the generaltheoryof relativity,the law of the con

stancy of the velocityof lightin vacuo, which consti

tutes one of the two fundamental assumptions in the

specialtheory of relativityand to which we have

alreadyfrequentlyreferred,cannot claim any unlimited

validity.A curvature of rays of lightcan only take

place when the velocityof propagation of lightvaries

with position. Now we might think that as a conse

quence of this,the specialtheoryof relativityand with

it the whole theory of relativitywould be laid in the

dust. But in realitythis is not the case. We can only

conclude that the specialtheory of relativitycannot

claim an unlimited domain of validity; its results

hold only so long as we are able to disregardthe in

fluences of gravitationalfields on the phenomena

(e.g.of light).Since it has often been contended by opponents of

the theory of relativitythat the specialtheory of

relativityis overthrown by the general theory of"rela

tivity,it is perhaps advisable to make the facts of the

case clearer by means of an appropriate comparison.Before the development of electrodynamicsthe laws

of electrostatics were looked upon as the laws of

electricity.At the present time we know that

electric fields can be derived correctlyfrom electro

static considerations only for the case, which is never

strictlyrealised,in which the electricalmasses are quiteat rest relativelyto each other, and to the co-ordinate

system. Should we be justifiedin sayingthat for this

Societies,the existence of the deflection of lightdemanded by

theory was confirmed duringthe solar eclipseof 20th Mav IQIQ

(Cf.Appendix III.)

INFERENCES FROM RELATIVITY 77

reason electrostatics is overthrown by the field-equa

tions of Maxwell in electrodynamics? Not in the least.

Electrostatics is contained in electrodynamicsas a

limitingcase ; the laws of the latter lead directlyto

those of the former for the case in which the fields are

invariable with regard to time. No fairer destiny

could be allotted to any physicaltheory,than that it

should of itself point out the way to the introduction

of a more comprehensivetheory, in which it lives on

as a limitingcase.

In the example of the transmission of lightjustdealt

with, we have seen that the generaltheory of relativity

enables us to derive theoreticallythe influence of a

gravitationalfield on the course of natural processes,

the laws of which are alreadyknown when a gravitational field is absent. But the most attractive problem,to the solution of which the generaltheory of relativity

suppliesthe key, concerns the investigationof the laws

satisfiedby the gravitationalfielditself. Let us consider

this for a moment.

We are acquaintedwith space-timedomains which

behave (approximately)in a" Galileian " fashion under

suitable choice of reference-body,i.e.domains in which

gravitationalfields are absent. If we nov -efer such

a domain to a reference-bodyK' possessingany kind

of motion, then relative to K' there exists a gravitational field which is variable with respect to space and

time.1 The character of this field will of course depend

on the motion chosen for K'. According to the general

theory of relativity,the generallaw oi the gravitationalfield must be satisfiedfor all gravitationalfields obtain-

1 This follows from a generalisationof the discussion in Section

XX.


able in this way.Even though by no means all gravita

tional fields canbe produced in this

way, yet we may

entertain the hope that the general law of gravitation

will be derivable from such gravitational fields of a

special kind. This hope has been realised in the most

beautiful manner. But between the clear vision of

this goal and its actual realisation it was necessary to

surmount a serious difficulty, and as this lies deep at

the root of things, I dare not withhold it from the reader.

We require to extend our ideas of the space-time con

tinuum still farther.

XXIII

BEHAVIOUR OF CLOCKS AND MEASURING-RODS

ON A ROTATING BODY OF REFERENCE

HITHERTOI have purposely refrained from

speaking about the physical interpretation of

space- and time-data in the case of the general

theory of relativity. As a consequence, I am guilty of a

certain slovenliness of treatment, which, as we know

from the special theory of relativity,is far from being

unimportant and pardonable. It is now high time that

we remedy this defect; but I would mention at the

outset, that this matter lays no small claims on the

patience and on the power of abstraction of the reader.

We start off again from quite special cases, which we

have frequently used before. Let us consider a space-

time domain in which no gravitational field exists

relative to a reference-body K whose state of motion

has been suitably chosen. K is then a Galileian refer

ence-body as regards the domain considered, and the

results of the special theory of relativity hold relative

to K. Let us suppose the same domain referred to a

second body of reference K', which is rotating uniformly

with respect to K. In order to fix our ideas, we shall

imagine K' to be in the form of a plane circular disc,

which rotates uniformly in its own plane about its

centre. An observer who is sitting eccentrically on the

79


disc K' is sensible of a force which acts outwards in a

radial direction,and which would be interpretedas an

effect of inertia (centrifugalforce)by an observer who

was at rest with respectto the originalreference-bodyK.

But the observer on the disc may regard his disc as a

reference-bodywhich is "at rest"

; on the basis of the

generalprincipleof relativityhe is justifiedin doingthis.

The force acting on himself, and in fact on all other

bodies which are at rest relative to the disc,he regards

as the effect of a gravitationalfield. Nevertheless,

the space-distributionof this gravitationalfield is of a

kind that would not be possibleon Newton's theory of

gravitation.1But since the observer believes in the

generaltheory of relativity,this does not disturb him ;

he is quitein the rightwhen he believes that a generallaw of gravitationcan be formulated" a law which not

only explainsthe motion of the stars correctly,but

also the field of force experiencedby himself.

The observer performs experiments on his circular

disc with clocks and measuring-rods. In doing so, it

is his intention to arrive at exact definitions for the

significationof time- and space-data with reference

to the circular disc K', these definitions being based on

his observations. What will be his experiencein this

enterprise?

To start with, he places one of two identicallyconstructed clocks at the centre of the circular disc,and the

'

other on the edge of the disc,so that they are at rest

relative to it. We now ask ourselves whether both

clocks go at the same rate from the standpoint of the

1 The field disappearsat the centre of the disc and increases

proportionallyto the distance from the centre as we proceedoutwards.

BEHAVIOUR OF CLOCKS AND RODS 81

non-rotatingGalileian reference-bodyK. As judgedfrom this body, the clock at the centre of the disc has

no velocity,whereas the clock at the edge of the disc

is in motion relative to K in consequence of the rotation.

Accordingto a result obtained in Section XII, it follows

that the latter clock goes at a rate permanentlyslower

than that of the clock at the centre of the circular disc,

i.e.as observed from K. It is obvious that the same effect

would be noted by an observer whom we will imagine

sittingalongsidehis clock at the centre of the circular

disc. Thus on our circular disc,or, to make the case

more general,in every gravitationalfield,a clock will

go more quicklyor less quickly,accordingto the positionin which the clock is situated (atrest). For this reason

it is not possibleto obtain a reasonable definition of time

with the aid of clocks which are arranged at rest with

respect to the body of reference. A similar difficulty

presents itself when we attempt to apply our earlier

definition of simultaneityin such a case, but I do not

wish to go any farther into this question.

Moreover, at this stage the definition of the space

co-ordinates also presents insurmountable difficulties.

If the observer applieshis standard measuring-rod

(a rod which is short as compared with the radius of

the disc)tangentiallyto the edge of the disc,then, as

judgedfrom the Galileian system, the lengthof this rod

will be lessthan I,since,accordingto Section XII, movingbodies suffer a shorteningin the direction of the motion.

On the other hand, the measuring-rodwill not experience

a shorteningin length,as judgedfrom K, if it isapplied

to the disc in the direction of the radius. If,then, the

observer first measures the circumference of the disc

with his measuring-rodand then the diameter of the

6


disc, on dividing the one by the other, he will not obtain

as quotient the familiar number 77=3-14. . .,

but

a larger number,1 whereas of course, for a disc which is

at rest with respect to K, this operation would yield IT

exactly. This proves that the propositions of Euclidean

geometry cannot hold exactly on the rotating disc, nor

in general in a gravitational field, at least if we attribute

the length i to the rod in all positions and in every

orientation. Hence the idea of a straight line also loses

its meaning. We are therefore not in a position to

define exactly the co-ordinates x, y, z relative to the

disc by means of the method used in discussing the

special theory, and as long as the co-ordinates and times

of events have not been defined, we cannot assign an

exact meaning to the natural laws in which these occur.

Thus all our previous conclusions based on general

relativity would appear to be called in question. In

reality we must make a subtle detour in order to be

able to apply the postulate of general relativity ex

actly. I shall prepare the reader for this in the

following paragraphs.

1 Throughout this consideration we have to use the Galileian

(non-rotating) system K as reference-body, since we may only

assume the validity of the results of the special theory of rela

tivity relative to K (relative to K' a gravitational field prevails).

XXIV

EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM

THE surface of a marble table is spread out in front

of me. I can get from any one point on this

table to any other point by passing continuously

from one point to a"

neighbouring"

one, and repeating

this process a (large) number of times, or, in other words,

by going from point to point without executing"

jumps."

I am sure the reader will appreciate with sufficient

clearness what I mean here by"

neighbouring"

and by"

jumps"

(ifhe is not too pedantic). We express this

property of the surface by describing the latter as a

continuum.

Let us now imagine that a large number of little rods

of equal length have been made, their lengths being

small compared with the dimensions of the marble

slab. When I say they are of equal length, I mean that

one can be laid on any other without the ends over

lapping. We next lay four of these little rods on the

marble slab so that they constitute a quadrilateral

figure (a square), the diagonals of which are equally

long. To ensure the equality of the diagonals, we make

use of a little testing-rod. To this square we add

similar ones, each of which has one rod in common

with the first. We proceed in like manner with each of

these squares until finally the whole marble slab is

83


laid out with squares. The arrangement is such, that

each side of a square belongsto two squares and each

corner to four squares.

It is a veritable wonder that we can carry out this

business without gettinginto the greatest difficulties.

We only need to think of the following. If at any

moment three squares meet at a corner, then two sides

of the fourth square are already laid, and, as a conse

quence, the arrangement of the remaining two sides of

the square is already completely determined. But I

am now no longer able to adjustthe quadrilateralsothat its diagonals may be equal. If they are equalof their own accord, then this is an especialfavour

of the marble slab and of the littlerods, about which I

can only be thankfullysurprised.We must needs

experiencemany such surprisesif the construction is to

be successful.

If everything has reallygone smoothly, then I say

that the pointsof the marble slab constitute a Euclidean

continuum with respectto the littlerod, which has been

used as a" distance

"

(line-interval).By choosingone corner of a square as

" origin,"I can characterise

every other corner of a square with reference to this

originby means of two numbers. I only need state

how many rods I must pass over when, startingfrom the

origin,I proceed towards the "

right" and then "

up

wards," in order to arrive at the corner of the square

under consideration. These two numbers are then the

" Cartesian co-ordinates "

of this corner with reference

to the " Cartesian co-ordinate system" which is deter

mined by the arrangement of littlerods.

By making use of the followingmodification of this

abstract experiment, we recognisethat there must also

EUCLIDEAN AND NON-EUCLIDEAN 85

be cases in which the experiment would be unsuccessful.

We shall suppose that the rods "

expand"

by an amount

proportionalto the increase of temperature. We heat

the central part of the marble slab,but not the peri

phery, in which case two of our little rods can still be

brought into coincidence at every positionon the table.

But our construction of squares must necessarilycomeinto disorder duringthe heating,because the littlerods

on the central region of the table expand, whereas

those on the outer part do not.

With reference to our little rods " denned as unit

lengths" the marble slab is no longer a Euclidean con

tinuum, and we are also no longerin the positionof de

nning Cartesian co-ordinates directlywith their aid,

since the above construction can no longer be carried

out. But since there are other thingswhich are not

influenced in a similar manner to the little rods (or

perhaps not at all)by the temperature of the table,it is

possiblequitenaturallyto maintain the point of view

that the marble slab is a" Euclidean continuum."

This can be done in a satisfactorymanner by making a

more subtle stipulationabout the measurement or the

comparison of lengths.But if rods of every kind (i.e.of every material)were

to behave in the same way as regards the influence of

temperature when they are on the variablyheated

marble slab, and if we had no other means of detecting

the effect of temperature than the geometricalbe

haviour of our rods in experimentsanalogous to the one

described above, then our best plan would be to assign

the distance one to two pointson the slab,providedthat

the ends of one of our rods could be made to coincide

with these two points; for how else should we define


the distance without our proceeding being in the highest

measure grossly arbitrary ? The method of Cartesian

co-ordinates must then be discarded, and replaced by

another which does not assume the validity of Euclidean

geometry for rigid bodies.1 The reader will notice that

the situation depicted here corresponds to the one

brought about by the general postulate of relativity

(Section XXIII).'

1 Mathematicians have been confronted with our problem in the

following form. If we are given a surface (e.g. an ellipsoid) in

Euclidean three-dimensional space, then there exists for this

surface a two-dimensional geometry, just as much as for a plane

surface. Gauss undertook the task of treating this two-dimen

sional geometry from first principles, without making use of the

fact that the surface belongs to a Euclidean continuum of

three dimensions. If we imagine constructions to be made with

rigid rods in the surface (similar to that above with the marble

slab), we should find that different laws hold for these from those

resulting on the basis of Euclidean plane geometry. The surface

is not a Euclidean continuum with respect to the rods, and we

cannot define Cartesian co-ordinates in the surface. Gauss

indicated the principles according to which we can treat the

geometrical relationships in the surface, and thus pointed out

the way to the method of Riemann of treating multi-dimen

sional, non-Euclidean conlinua. Thus it is that mathematicians

long ago solved the formal problems to which we are led by the

general postulate of relativity.

XXV

GAUSSIAN CO-ORDINATES

ACCORDINGto Gauss, this combined analytical

and geometrical mode of handling the problem

can be arrived at in the following way. We

imagine a system of arbitrary curves (see Fig. 4)

drawn on the surface of the table. These we desig

nate as w-curves, and we indicate each of them by

means of a number. The curves "=i, u=2 and

w=3 are drawn in the diagram. Between the curves

w=i and "=2 we must imagine an infinitely large

number to be drawn, all of which correspond to real

numbers lying between I and 2. We have then

a system of w-curves, and

this "infinitely dense" sys

tem covers the whole sur

face of the table. These

w-curves must not intersect

each other, and through each

point of the surface one and

only one curve must pass.

Thus a perfectly definite

value of u belongs to every point on the surface of the

marble slab. In like manner we imagine a system of

w-curves drawn on the surface. These satisfy the same

conditions as the w-curves, they are provided with num-

87

FIG. 4.


bers in a correspondingmanner, and they may likewise

be of arbitraryshape. It follows that a value of u and

a value of v belong to every pointon the surface of the

table. We call these two numbers the co-ordinates

of the surface of the table (Gaussianco-ordinates).For example, the pointP in the diagramhas the Gaussian

co-ordinates "=3, v=i. Two neighbouringpoints P

and P' on the surface then correspondto the co-ordinates

P: u,v

P' : u-\-du,v-\-dv,

where du and dv signifyvery small numbers. In a

similar manner we may indicate the distance (line-

interval)between P and P', as measured with a little

rod, by means of the very small number ^s. Then

accordingto Gauss we have

where gn, g12, g22, are magnitudes which depend in a

perfectlydefinite way on u and v. The magnitudes gu,

g12 and g22 determine the behaviour of the rods relative

to the w-curves and u-curves, and thus also relative

to the surface of the table. For the case in which the

pointsof the surface considered form a Euclidean con

tinuum with reference to the measuring-rods,but

only in this case, it is possibleto draw the w-curves

and 5y-curves and to attach numbers to them, in such a

manner, that we simplyhave :

ds2=du*+dv*.

Under these conditions, the w-curves and v-curves are

straightlines in the sense of Euclidean geometry, and

they are perpendicularto each other. Here the Gaussian

co-ordinates are simply Cartesian ones. It is clear

GAUSSIAN CO-ORDINATES 89

that Gauss co-ordinates are nothing more than an

association of two sets of numbers with the points of

the surface considered,of such a nature that numerical

values differingvery slightlyfrom each other are

associated with neighbouringpoints" in space."

So far, these considerations hold for a continuum

of two dimensions. But the Gaussian method can be

applied also to a continuum of three, four or more

dimensions. If, for instance, a continuum of four

dimensions be supposed available,we may represent

it in the followingway. With every point of the

continuum we associate arbitrarilyfour numbers, xv x2,

X3, #4, which are known as" co-ordinates." Adjacent

pointscorrespondto adjacentvalues of the co-ordinates.

If a distance ^s is associated with the adjacent pointsP and P', this distance being measurable and well-

defined from a physicalpointof view, then the following

formula holds :

where the magnitudes gu, etc.,have values which vary

with the positionin the continuum. Only when the

continuum is a Euclidean one is it possibleto associate

the co-ordinates xl . . #4 with the points of the

continuum so that we have simply

In this case relations hold in the four-dimensional

continuum which are analogous to those holdingin our

three-dimensional measurements.

However, the Gauss treatment for dsz which we have

given above is not always possible.It is only possible

when sufficientlysmall regionsof the continuum under

consideration may be regardedas Euclidean continua.

For example, this obviously holds in the case of the

marble slab of the table and local variation of temperature.

The temperature is practically constant for a small

part of the slab, and thus the geometrical behaviour of

the rods is almost as it ought to be according to the

rules of Euclidean geometry. Hence the imperfections

of the construction of squares in the previous section

do not show themselves clearly until this construction

is extended over a considerable portion of the surface

of the table.

We can sum this up as follows : Gauss invented a

method for the mathematical treatment of continua in

general, in which " size-relations "

(" distances " between

neighbouring points) are defined. To every point of a

continuum are assigned as many numbers (Gaussian co

ordinates) as the continuum has dimensions. This is

done in such a way, that only one meaning can be attached

to the assignment, and that numbers (Gaussian co

ordinates) which differ by an indefinitely small amount

are assigned to adjacent points. The Gaussian co

ordinate system is a logical generalisation of the Cartesian

co-ordinate system. It is also applicable to non-Euclidean

continua, but only when, with respect to the defined

" size "

or"

distance," small parts of the continuum

under consideration behave more nearly like a Euclidean

system, the smaller the part of the continuum under

our notice.

XXVI

THE SPACE-TIME CONTINUUM OF THE SPECIAL

THEORY OF RELATIVITY CONSIDERED AS

A EUCLIDEAN CONTINUUM

WE are now in a position to formulate more

exactly the idea of Minkowski, which was

only vaguely indicated in Section XVII.

,In accordance with the special theory of relativity,

certain co-ordinate systems are given preference

for the description of the four-dimensional, space-time

continuum. We called these " Galileian co-ordinate

systems." For these systems, the four co-ordinates

x, y, z, t, which determine an event or "

in other

words" a point of the four-dimensional continuum, are

defined physically in a simple manner, as set forth in

detail in the first part of this book. For the transition

from one Galileian system to another, which is moving

uniformly with reference to the first, the equations of

the Lorentz transformation are valid. These last

form the basis for the derivation of deductions from the

special theory of relativity, and in themselves they are

nothing more than the expression of the universal

validity of the law of transmission of light for all Galileian

systems of reference.

Minkowski found that the Lorentz transformations

satisfy the following simple conditions. Let us consider

91


two neighbouringevents, the relative positionof which

in the four-dimensional continuum is given with respectto a Galileian reference-bodyK by the space co-ordinate

differences dx, dy, dz and the time-difference dt. With

reference to a second Galileian system we shall suppose

that the correspondingdifferences for these two events

are dx',dy',dz',dt'. Then these magnitudes alwaysfulfilthe condition x

The validityof the Lorentz transformation follows

from this condition. We can express this as follows :

The magnitude

which belongs to two adjacent points of the four-

dimensional space-timecontinuum, has the same value

for all selected (Galileian)reference-bodies. If we re

placex, y, z, "J- i ct,by xit x2, xs, #4, we also obtain the

result that

is independentof the choice of the body of reference.

We call the magnitude ds the " distance "

apart of the

two events or four-dimensional points.

Thus, if we choose as time-variable the imaginary

variable \l- 1 ct instead of the real quantityt,we can

regard the space-time continuum " in accordance with

the specialtheory of relativity" as a" Euclidean "

four-dimensional continuum, a result which follows

from the considerations of the precedingsection.

1 Cf. Appendices I and II. The relations which are derived

there for the co-ordinates themselves are valid also for co

ordinate differences,and thus also for co-ordinate differentials

(indefinitelysmall differences).

XXVII

THE SPACE -TIME CONTINUUM OF THE

GENERAL THEORY OF RELATIVITY IS

NOT A EUCLIDEAN CONTINUUM

INthe first part of this book we were able to make use

of space-time co-ordinates which allowed of a simple

and direct physical interpretation, and which, accord

ing to Section XXVI, can be regarded as four-dimensional

Cartesian co-ordinates. This was possible on the basis

of the law of the constancy of the velocity of light. But

according to Section XXI, the general theory of relativity

cannot retain this law. On the contrary, we arrived at

the result that according to this latter theory the

velocity of light must always depend on the co-ordinates

when a gravitational field is present. In connection

with a specific illustration in Section XXIII, we found

that the presence of a gravitational field invalidates the

definition of the co-ordinates and the time, which led us

to our objective in the special theory of relativity.

In view of the results of these considerations we are

led to the conviction that, according to the general

principle of relativity, the space-time continuum cannot

be regarded as a Euclidean one, but that here we have

the general case, corresponding to the marble slab with

local variations of temperature, and with which we

made acquaintance as an example of a two-dimensional

93

continuum. Just as it was there impossibleto construct

a Cartesian co-ordinate system from equal rods, so

here it is impossibleto build up a system (reference-

body) from rigidbodies and clocks, which shall be of

such a nature that measuring-rods and clocks, arranged

rigidlywith respect to one another, shall indicate position and time directly. Such was the essence of the

difficultywith which we were confronted in Section

XXIII.

But the considerations of Sections XXV and XXVI

show us the way to surmount this difficulty.We refer the

four-dimensional space-time continuum in an arbitrary

manner to Gauss co-ordinates. We assign to every

point of the continuum (event) four numbers, xlt x2,

#3, x" (co-ordinates),which have not the least direct

physical significance,but only serve the purpose of

numbering the points of the continuum in a definite

but arbitrarymanner. This arrangement does not even

need to be of such a kind that we must regardxlt x2, x3 as

"

space" co-ordinates and #4 as a

" time " co-ordinate.

The reader may think that such a descriptionof the

world would be quiteinadequate. What does it mean

to assign to an event the particular co-ordinates xlf

xt" xs" x"" if m themselves these co-ordinates have no

significance? More careful consideration shows, how

ever, that this anxiety is unfounded. Let us consider,

for instance, a material point with any kind of motion.

If this point had only a momentary existence without

duration, then it would be described in space-timeby a

singlesystem of values xitx2,x3,x^ Thus its permanentexistence must be characterised by an infinitelylargenumber of such systems of values, the co-ordinate values

of which are so close together as to give continuity ;

SPACE-TIME CONTINUUM 95

correspondingto the material point, we thus have a

(uni-dimensional)line in the four-dimensional continuum.

In the same way, any such lines in our continuum

correspond to many pointsin motion. The only state

ments having regard to these points which can claim

a physicalexistence are in realitythe statements about

their encounters. In our mathematical treatment,

such an encounter is expressedin the fact that the two

lines which represent the motions of the points in

questionhave a particularsystem of co-ordinate values,

xi" xz" X3" Xv m common. After mature consideration

the reader will doubtless admit that in realitysuch

encounters constitute the only actual evidence of a

time-space nature with which we meet in physicalstatements.

When we were describingthe motion of a material

point relative to a body of reference, we stated

nothing more than the encounters of this point with

particularpoints of the reference-body.We can also

determine the correspondingvalues of the time by the

observation of encounters of the body with clocks, in

conjunctionwith the observation of the encounter of the

hands of clocks with particularpoints on the dials.

It is justthe same in the case of space-measurements by

means of measuring-rods,as a little consideration will

show.

The followingstatements hold generally: Every

physical descriptionresolves itself into a number of

statements, each of which refers to the space-timecoincidence of two events A and B. In terms of

Gaussian co-ordinates,every such statement isexpressed

by the agreement of their four co-ordinates xlt x2, x3,

%4. Thus in reality,the descriptionof the time-space


continuum by meansof Gauss co-ordinates completely

replaces the description with the aid of a body ofre

ference, without suffering from the defects of the latter

'mode of description ; it is not tied down to the Euclidean

character of the continuum which has to be represented.

XXVIII

EXACT FORMULATION OF THE GENERAL


WE are now in a position to replace the pro

visional formulation of the general principle

of relativity given in Section XVIII by

an exact formulation. The form there used, "All

bodies of reference K, K',

etc., are equivalent for

the description of natural phenomena (formulation of

the general laws of nature), whatever may be their

state of motion," cannot be maintained, because the

use of rigid reference-bodies, in the sense of the method

followed in the special theory of relativity,is in gereral

not possible in space-time description. The Gauss

co-ordinate system has to take the place of the body of

reference. The following statement corresponds to the

fundamental idea of the general principle of relativity:

" All Gaussian co-ordinate systems are essentially equi

valent for the formulation of the general laws of nature."

We can state this general principle of relativityin still

another form, which renders it yet more clearly in

telligiblethan it is when in the form of the natural

extension of the special principle of relativity. Accord

ing to the special theory of relativity,the equations

which express the general laws of nature pass over into

equations of the same form when, by making use of the

Lorentz transformation, we replace the space-time

7


variables x, y, z, t, of a (Galileian)reference-bodyK

by the space-timevariables x',y',z',t',of a new re

ference-body K'. According to the general theoryof relativity,on the other hand, by applicationof

arbitrarysubstitutions of the Gauss variables xl} x2, % x",

the equationsmust pass over into equationsof the same

form ; for every transformation (not only the Lorentz

transformation)corresponds to the transition of one

Gauss co-ordinate system into another.

If we desire to adhere to our" old-time " three-

dimensional view of things,then we can characterise

the development which is being undergone by the

fundamental idea of the general theory of relativity

as follows : The specialtheory of relativityhas reference

to Galileian domains, i.e.to those in which no gravita

tional field exists. In this connection a Galileian re

ference-body serves as body of reference,i.e. a rigid

body the state of motion of which is so chosen that the

Galileian law of the uniform rectilinear motion of

" isolated " material pointsholds relativelyto it.

Certain considerations suggest that we should refer

the same Galileian domains to non-Galileian reference-

bodies also. A gravitationalfield of a specialkind is

then presentwith respectto these bodies (cf.Sections XX

and XXIII).In gravitationalfields there are no such thingsas rigid

bodies with Euclidean properties; thus the fictitiousrigid

body of reference is of no avail in the generaltheory of

relativity.The motion of clocks is also influenced by

gravitationalfields,and in such a way that a physicaldefinition of time which is made directlywith the aid of

clocks has by no means the same degree of plausibilityas in the specialtheory of relativity.

GENERAL PRINCIPLE OF RELATIVITY 99

For this reason non-rigidreference-bodies are used,

which are as a whole not only moving in any way

whatsoever, but which also suffer alterations in form

ad lib.during their motion. Clocks, for which the law of

motion is of any kind, however irregular,serve for the

definition of time. We have to imagine each of these

clocks fixed at a point on the non-rigidreference-body.These clocks satisfyonly the one condition, that the

"

readings" which are observed simultaneously on

adjacent clocks (inspace) differ from each other by an

indefinitelysmall amount. This non-rigid reference-

body, which might appropriatelybe termed a" reference-

mollusk," is in the main equivalentto a Gaussian four-

dimensional co-ordinate system chosen arbitrarily.That which gives the "mollusk" a certain compre-

hensibleness as compared with the Gauss co-ordinate

system is the (reallyunjustified) formal retention of

the separate existence of the space co-ordinates as

opposed to the time co-ordinate. Every point on the

mollusk is treated as a space-point,and every material

point which is at rest relativelyto it as at rest, so long as

the mollusk is considered as reference-body. The

general principleof relativityrequires that all these

mollusks can be used as reference-bodies with equal

rightand equal success in the formulation of the general

laws of nature ; the laws themselves must be quite

independent of the choice of mollusk.

The great power possessed by the general principle

of relativitylies in the comprehensive limitation which

is imposed on the laws of nature in consequence of what

we have seen above.

XXIX

THE SOLUTION OF THE PROBLEM OF GRAVI

TATION ON THE BASIS OF THE GENERAL


IFthe reader has followed all our previous con

siderations, he will have no further difficulty in

understanding the methods leading to the solution

of the problem of gravitation.

We start off from a consideration of a Galileian

domain, i.e. a domain in which there is no gravitational

field relative to the Galileian reference-body K. The

behaviour of measuring-rods and clocks with reference

to K is known from the special theory of relativity,

likewise the behaviour of " isolated " material points ;

the latter move uniformly and in straight lines.

Now let us refer this domain to a random Gauss co

ordinate system or to a" mollusk "

as reference-body

K'. Then with respect to K' there is a gravitational

field G (of a particular kind). We learn the behaviour

of measuring-rods and clocks and also of freely-moving

material points with reference to K' simply by mathe

matical transformation. We interpret this behaviour

as the behaviour of measuring-rods, clocks and material

points under the influence of the gravitational field G.

Hereupon we introduce a hypothesis : that the in

fluence of the gravitational field on measuring-rods,

SOLUTION OF GRAVITATION 101

clocks and freely-movingmaterial points continues to

take placeaccordingrtothe same laws, even in the case

when the prevailinggravitationalfield is not derivable

from the Galileian specialcase, simply by means of a

transformation of co-ordinates.

The next step is to investigatethe space-timebehaviour of the gravitationalfieldG, which was derived

from the Galileian specialcase simplyby transformation

of the co-ordinates. This behaviour is formulated

in a law, which is always valid, no matter how the

reference-body(mollusk)used in the descriptionmaybe chosen.

This law is not yet the generallaw of the gravitationalfield,since the gravitationalfield under consideration is

of a specialkind. In order to find out the generallaw-of-field of gravitationwe still requireto obtain a

generalisationof the law as found above. This can be

obtained without caprice,however, by taking into

consideration the followingdemands :

(a)The requiredgeneralisationmust likewise satisfythe generalpostulateof relativity.

(6)If there is any matter in the domain under con

sideration, only its inertia! mass, and thus

accordingto Section XV only its energy is of

importance for its effect in excitinga field.

(c)Gravitational field and matter together must

satisfythe law of the conservation of energy

(and of impulse).

Finally,the generalprincipleof relativitypermits

us to determine the influence of the gravitationalfield

on the course of all those processes which take place

accordingto known laws when a gravitationalfield is


absent, i.e. which have already been fitted into the

frame of the specialtheory of relativity.In this con

nection we proceed in principleaccordingto the method

which has alreadybeen explained for measuring-rods,clocks and freely-movingmaterial points.

The theory of gravitationderived in this way from

the general postulateof relativityexcels not only in

its beauty ; nor in removing the defect attaching to

classical mechanics which was brought to lightin Section

XXI ; nor in interpretingthe empiricallaw of the equalityof inertial and gravitationalmass ; but it has also

alreadyexplained a result of observation in astronomy,

againstwhich classical mechanics is powerless.If we confine the applicationof the theory to the

case where the gravitationalfields can be regarded as

beingweak, and in which all masses move with respectto the co-ordinate system with velocities which are

small compared with the velocityof light,we then obtain

as a first approximation the Newtonian theory. Thus

the latter theoryis obtained here without any particular

assumption, whereas Newton had to introduce the

hypothesisthat the force of attraction between mutuallyattractingmaterial points is inverselyproportionaltothe square of the distance between them. If we in

crease the accuracy of the calculation,deviations from

the theory of Newton make their appearance, practi

callyall of which must nevertheless escape the test of

observation owing to their smallness.

We must draw attention here to one of these devia

tions. According to Newton's theory,a planet moves

round the sun in an ellipse,which would permanentlymaintain its positionwith respect to the fixed stars,

if we could disregardthe motion of the fixed stars

SOLUTION OF GRAVITATION 103

themselves and the action of the other planetsunderconsideration. Thus, if we correct the observed motion

of the planetsfor these two influences,and if Newton's

theory be strictlycorrect, we ought to obtain for the

orbit of the planet an ellipse,which is fixed with re

ference to the fixed stars. This deduction, which can

be tested with great accuracy, has been confirmed

for all the planets save one, with the precisionthat is

capable of beingobtained by the delicacyof observation

attainable at the present time. The sole exceptionis Mercury,the planetwhich lies nearest the sun. Since

the time of Leverrier,it has been known that the ellipse

correspondingto the orbit of Mercury, after it has been

corrected for the influences mentioned above, is not

stationarywith respect to the fixed stars, but that it

rotates exceedinglyslowly in the plane of the orbit

and in the sense of the orbital motion. The value

obtained for this rotary movement of the orbital ellipse

was 43 seconds of arc per century, an amount ensured

to be correct to within a few seconds of arc. This

effect can be explainedby means of classical mechanics

only on the assumption of hypotheses which have

little probability,and which were devised solelyfor

this purpose.

On the basis of the generaltheory of relativity,it

is found that the ellipseof every planetround the sun

must necessarilyrotate in the manner indicated above ;

that for all the planets,with the exceptionof Mercury,

this rotation is too small to be detected with the delicacy

of observation possibleat the present time ; but that in

the case of Mercury it must amount to 43 seconds of

arc per century, a result which is strictlyin agreement

with observation.


Apart from this one, it has hitherto been possible to

make only two deductions from the theory which admit

of being tested by observation, to wit, the curvature

of light rays by the gravitational field of the sun,1

and a displacement of the spectral lines of light reaching

us from large stars, as compared with the corresponding

lines for light produced in an analogous manner terres

trially (i.e. by the same kind of molecule). I do not

doubt that these deductions from the theory will be

confirmed also.

1 Observed by Eddington and others in 1919. (Cf. Appendix

III.)

PART III

CONSIDERATIONS ON THE UNIVERSE AS

A WHOLE

XXX

COSMOLOGICAL DIFFICULTIES OF NEWTON'S

THEORY

APART from the difficulty discussed in Section

XXI, there is a second fundamental difficulty

attending classical celestial mechanics, which,

to the best of my knowledge, was first discussed in

detail by the astronomer Seeliger. If we ponder over

the question as to how the universe, considered as a

whole, is to be regarded, the first answer that suggests

itself to us is surely this : As regards space (and time)

the universe is infinite. There are stars everywhere,

so that the density of matter, although very variable

in detail, is nevertheless on the average everywhere the

same. In other words : However far we might travel

through space, we should find everywhere an attenuated

swarm of fixed stars of approximately the same kind

and density.

This view is not in harmony with the theory of

Newton. The latter theory rather requires that the

universe should have a kind of centre in which the

105

106 CONSIDERATIONS ON THE UNIVERSE

densityof the stars is a maximum, and that as we

proceed outwards from this centre the group-densityof the stars should diminish, until finally,at great

distances,it is succeeded by an infinite regionof emptiness. The stellar universe ought to be a finite island in

the infinite ocean of space.1This conception is in itself not very satisfactory.

It is stillless satisfactorybecause it leads to the result

that the lightemitted by the stars and also individual

stars of the stellar system are perpetuallypassingout

into infinite space, never to return, and without ever

again coming into interaction with other objectsof

nature. Such a finite material universe would be

destined to become graduallybut systematicallyim

poverished.In order to escape this dilemma, Seeligersuggesteda

modification of Newton's law, in which he assumes that

for great distances the force of attraction between two

masses diminishes more rapidlythan would result from

the inverse square law. In this way it is possiblefor the

mean density of matter to be constant everywhere,even to infinity,without infinitelylargegravitationalfields being produced. We thus free ourselves from the

1 Proof " According to the theory of Newton, the number of

" lines of force " which come from infinityand terminate in a

mass m is proportionalto the mass m. If,on the average, the

mass-density p0 is constant throughout the universe, then a

sphere of volume V will enclose the average mass pQV. Thus

the number of lines of force passingthrough the surface F of the

sphere into its interior is proportionalto p0V. For unit area

of the surface of the sphere the number of lines of force which

enters the sphere is thus proportionalto /"0" or to /"0R. HenceF

the intensityof the field at the surface would ultimatelybecome

infinite with increasingradius R of the sphere,which is impossible.

DIFFICULTIES OF NEWTON'S THEORY 107

distasteful conception that the material universe ought

to possess something of the nature ofa centre. Of

course we purchase our emancipation from the funda

mental difficulties mentioned, at the cost of a modifica

tion and complication of Newton's law which has

neither empirical nor theoretical foundation. We can

imagine innumerable laws which would serve the same

purpose,without our being able to state a reason why

oneof them is to be preferred to the others ; for

any

oneof these laws would be founded just as little

on

more general theoretical principles as is the law of

Newton.

XXXI

THE POSSIBILITY OF A "FINITE" AND YET

"UNBOUNDED" UNIVERSE

BUT speculations on the structure of the universe

also move in quite another direction. The

development of non-Euclidean geometry led to

the recognition of the fact, that we can cast doubt on the

infiniteness of our space without coming into conflict

with the laws of thought or with experience (Riemann,

Helmholtz).

These questions have already been treated

in detail and with unsurpassable lucidity by Helm

holtz and Poincare, whereas I can only touch on them

briefly here.

In the first place, we imagine an existence in two-

dimensional space. Flat beings with flat implements,

and in particular flat rigid measuring-rods, are free to

move in a plane. For them nothing exists outside of

this plane: that which they observe to happen to

themselves and to their flat "

things" is the all-inclusive

reality of their plane. In particular, the constructions

of plane Euclidean geometry can be carried out by

means of the rods, e.g. the lattice construction, considered

in Section XXIV. In contrast to ours, the universe of

these beings is two-dimensional ; but, like ours, it extends

to infinity. In their universe there is room for an

infinite number of identical squares made up of rods

inR*

*

UNIVERSE" FINITE YET UNBOUNDED 109

i.e.its volume (surface)is infinite. If these beingssaytheir universe is "

plane," there is sense in the state

ment, because they mean that they can perform the con

structions of plane Euclidean geometry with their rods.

In this connection the individual rods always representthe same distance,independentlyof their position.

Let us consider now a second two-dimensional exist

ence, but this time on a sphericalsurface instead of on

a plane. The flat beings with their measuring-rodsand other objectsfit exactlyon this surface and they

are unable to leave it. Their whole universe of observa

tion extends exclusivelyover the surface of the sphere.Are these beingsable to regard the geometry of their

universe as beingplane geometry and their rods withal

as the realisation of " distance " ? They cannot do

this. For ifthey attempt to realise a straightline,theywill obtain a curve, which we

" three-dimensional

beings"

designateas a great circle,i.e.a self-contained

line of definite finite length,which can be measured

up by means of a measuring-rod. Similarly,this

universe has a finite area that can be compared with the

area of a square constructed with rods. The great

charm resultingfrom this consideration lies in the

recognitionof the fact that the universe ofthese beingsis

finiteand yet has no limits.

But the spherical-surfacebeings do not need to go

on a world-tour in order to perceivethat they are not

living in a Euclidean universe. They can convince

themselves of this on every part of their " world,"

providedtheydo not use too small a pieceof it. Starting

from a point,they draw"

straightlines"

(arcsof circles

as judged in three-dimensional space)of equal length

in all directions. They will call the line joiningthe


free ends of these lines a" circle." For a planesurface,

the ratio of the circumference of a circle to its diameter,

both lengthsbeing measured* with the same rod, is,

accordingto Euclidean geometry of the plane,equal to

a constant value TT, which is independentof the diameter

of the circle. On their sphericalsurface our flat beingswould find for this ratio the value

i.e. a smaller value than TT, the difference being the

more considerable, the greater is the radius of the

circle in comparison with the radius R of the " world-

sphere*" By means of this relation the sphericalbeings

can determine the radius of their universe ("world "),

even when only a relativelysmall part of their world-

sphere is available for their measurements. But if this

part is very small indeed, they will no longerbe able to

demonstrate that they are on a spherical" world " and

not on a Euclidean plane,for a small part of a sphericalsurface differs only slightlyfrom a piece of a planeof

the same size.

Thus if the spherical-surfacebeings are livingon a

planetof which the solar system occupiesonlya negligiblysmall part of the sphericaluniverse,they have no means

of determiningwhether they are livingin a finite or in

an innnite universe,because the "

piece of universe"

to which they have access is in both cases practicallyplane, or Euclidean. It follows directlyfrom this

discussion,that for our sphere-beingsthe circumference

of a circle firstincreases with the radius until the" cir-

UNIVERSE" FINITE YET UNBOUNDED 111

cumference of the universe " is reached, and that it

thenceforward gradually decreases to zero for still

further increasingvalues of the radius. During this

process the area of the circle continues to increase

more and more, until finallyit becomes equal to the

total area of the whole "

world-sphere."

Perhaps the reader will wonder why we have placedour

"

beings"

on a sphererather than on another closed

surface. But this choice has its justificationin the fact

that,of allclosed surfaces,the sphereisunique in possess

ingthe property that all pointson it are equivalent. I

admit that the ratio of the circumference c of a circle

to its radius r depends on r, but for a given value of r

it is the same for all points of the "

world-sphere"

;

in other words, the "

world-sphere"

is a" surface of

constant curvature."

To this two-dimensional sphere-universethere is a

three-dimensional analogy,namely,the three-dimensional

spheriqalspace which was discovered by Riemann. Its

pointsare likewise all equivalent. It possesses a finite

volume, which is determined by its "radius" (2,Tr2R3).Is it possibleto imagine a sphericalspace ? To imagine

a space means nothing else than that we imagine an

epitome of our"

space"

experience,i.e. of experience

that we can have in the movement of "

rigid" bodies.

In this sense we can imaginea sphericalspace.

Suppose we draw lines or stretch stringsin all direc

tions from a point,and mark off from each of these

the distance r with a measuring-rod. All the free end-

points of these lengthslieion a sphericalsurface. We

can speciallymeasure up the area (F) of this surface

by means of a%quare made up of measuring-rods.If

the universe is Euclidean, then F=q*rz ; ifitisspherical,


then F is always less than 47^. With increasingvalues of r, F increases from zero up to a maximum

value which is determined by the"

world-radius," but

for stillfurther increasingvalues of r, the area graduallydiminishes to zero. At first,the straightlines which

radiate from the startingpoint diverge farther and

farther from one another, but later they approach

each other, and finallythey run together again at a

"

counter-point"

to the startingpoint. Under such

conditions they have traversed the whole spherical

space. It is easilyseen that the three-dimensional

sphericalspace isquiteanalogousto the two-dimensional

sphericalsurface. It is finite (i.e.of finite volume),and

has no bounds.

It may be mentioned that there is yet another kind

of curved space :"

ellipticalspace." It can be regarded

as a curved space in which the two"

counter-points"

are identical (indistinguishablefrom each other). An

ellipticaluniverse can thus be considered to some

extent as a curved universe possessingcentral symmetry.It follows from what has been said,that closed spaces

without limits are conceivable. From amongst these,

the sphericalspace (and the elliptical)excels in its

simplicity,since all points on it are equivalent. As a

result of this discussion,a most interestingquestionarises for astronomers and physicists,and that is

whether the universe in which we live is infinite,or

whether it is finite in the manner of the sphericaluni

verse. Our experienceis far from being sufficient to

enable us to answer this question. But the general

theoryof relativitypermits of our answering it with a

moderate degreeof certainty,and in this connection the

difficultymentioned in Section XXX finds its solution.

XXXII

THE STRUCTURE OF SPACE ACCORDING TO


ACCORDINGto the general theory of relativity,the geometrical properties of space are not in

dependent, but they are determined by matter.

Thus we can draw conclusions about the geometrical

structure of the universe only if we base our considera

tions on the state of the matter as being something

that is known. We know from experience that, for a

suitably chosen co-ordinate system, the velocities of

the stars are small as compared with the velocity of

transmission of light. We can thus as a rough ap

proximation arrive at a conclusion as to the nature of

the universe as a whole, if we treat the matter as being

at rest.

We already know from our previous discussion that the

behaviour of measuring-rods and clocks is influenced by

gravitational fields, i.e. by the distribution of matter.

This in itself is sufficient to exclude the possibility of

the exact validity of Euclidean geometry in our uni

verse. But it is conceivable that our universe differs

only slightly from a Euclidean one, and this notion

seems all the more probable, since calculations show

that the metrics of surrounding space is influenced only

to an exceedingly small extent by masses even of the

8


magnitude of our sun. We might imagine that, as

regards geometry, our universe behaves analogously

to a surface which is irregularlycurved in its individual

parts, but which nowhere departs appreciablyfrom a

plane : something like the rippled surface of a lake.

Such a universe might fittinglybe called a quasi-

Euclidean universe. As regards its space it would be

infinite. But calculation shows that in a quasi-

Euclidean universe the average density of matter

would necessarilybe nil. Thus such a universe could

not be inhabited by matter everywhere ; it would

present to us that unsatisfactorypicture which we

portrayedin Section XXX.

If we are to have in the universe an average densityof matter which differs from zero, however small may

be that difference,then the universe cannot be quasi-Euclidean. On the contrary, the results of calculation

indicate that if matter be distributed uniformly, the

universe would necessarilybe spherical(or elliptical).Since in realitythe detailed distribution of matter is

not uniform, the real universe will deviate in individual

parts from the spherical,i.e.the universe will be quasi-

spherical.But it will be necessarilyfinite. In fact,the

theory supplies us with a simple connection x between

the space-expanse of the universe and the average

densityof matter in it.

1 For the " radius " JR of the universe we obtain the equation

The use of the C.G.S. system in this equation gives?= ro8.10";

is the average density of the matter.

APPENDIX I

SIMPLE DERIVATION OF THE LORENTZ

TRANSFORMATION [SUPPLEMENTARY TO SEC

TION XI]

FORthe relative orientation of the co-ordinate

systems indicated in Fig. 2, the #-axes of both

systems permanently coincide. In the present

case we can divide the problem into parts by considering

first only events which are localised on the #-axis. Any

such event is represented with respect to the co-ordinate

system K by the abscissa x and the time t, and with

respect to the system K' by the abscissa x' and the

time t'. We require to find x' and t' when x and t are

given.

A light-signal,which is proceeding along the positive

axis of x, is transmitted according to the equation

x"ct

or

X" Ct=Q. . .

(l).

Since the same light-signalhas to be transmitted relative

to K' with the velocity c, the propagation relative to

the system K' will be represented by the analogous

formula

x'"ct'=o...

(2)

Those space-time points (events) which satisfy (i) must

116 APPENDIX I

also satisfy(2). Obviouslythis will be the case when

the relation

(x'-cf)=X(x^c^ . . (3).

is fulfilledin general,where A. indicates a constant ; for,

according to (3),the disappearanceof (x" ct)involves

the disappearanceof (x'"ct'}.If we apply quitesimilar considerations to lightrays

which are being transmitted along the negative#-axis,

we obtain the condition

(x'+ct')=}*(x+ct)- " " (4)-

By adding (orsubtracting)equations(3)and (4),and

introducingfor convenience the constants a and b in

placeof the constants X and p., where

and

7.A. "

2

we obtain the equations

x' = ax-bct\ /-\

ct'= act - bxj

We should thus have the solution of our problem,if the constants a and b were known. These result

from the followingdiscussion.

For the originof K' we have permanentlyx'=o, and

hence accordingto the firstof the equations(5)

be~

a'

If we call v the velocitywith which the originof K' is

moving relative to K, we then have

be*--.,.. (6).


The same value v can be obtained from equation (5),if we calculate the velocityof another point of K'

relative to K, or the velocity(directedtowards the

negative#-axis)of a point of K with respect to K'. In

short, we can designatev as the relative velocityof the

two systems.

Furthermore, the principleof relativityteaches us

that, as judged from K, the length of a unit measuring-

rod which is at rest with reference to K' must be exactly

the same as the length,as judged from K', of a unit

measuring-rod which is at rest relative to K. In order

to see how the points of the #'-axis appear as viewed

from K, we only require to take a "snapshot" of K'

from K ; this means that we have to insert a particular

value of t (time of K), e.g. t=o. For this value of t

we then obtain from the firstof the equations(5)

x'=ax.

Two points of the #'-axis which are separatedby the

distance A#'=i when measured in the K' system are

thus separated in our instantaneous photograph by the

distance

A,-! . . .(7).

But if the snapshot be taken from K'(t'=o), and if

we eliminate t from the equations (5),taking into

account the expression(6),we obtain

From this we conclude that two pointson the #-axis

and separated by the distance i (relativeto K) will

be representedon our snapshot by the distance

118 APPENDIX I

But from what has been said, the two snapshotsmust be identical ; hence A* in (7) must be equal to

A*' in (jo),

so that we obtain

The equations (6)and (76)determine the constants a

and b. By insertingthe values of these constants in (5),

we obtain the first and the fourth of the equations

given in Section XI.

/ x - vt

/ u2

VI- -5V /"^

=

(8).

C,

Thus we have obtained the Lorentz transformation

for events on the #-axis. It satisfies the condition

The extension of this result,to include events which

take place outside the #-axis,is obtained by retaining

equations (8)and supplementingthem by the relations

":?} " " " " (9)-

In this way we satisfythe postulateof the constancy of

the velocityof lightin vacua for rays of lightof arbitrary

direction, both for the system K and for the system K',

This may be shown in the followingmanner.

We suppose a light-signalsent out from the originof K at the time "=o. It will be propagated according

to the equation


or, if we square this equation,accordingto the equation

ijr i . . \"J-

It is requiredby the law of propagation of light,inconjunctionwith the postulateof relativity,that the

transmission of the signalin questionshould take place"as judged from K'" in accordance with the corre

spondingformula

r'=cif,

or,

In order that equation (TOO)may be a consequence of

equation (10),we must have

(ii).

Since equation (8a) must hold for points on the

#-axis,we thus have o-=i. It is easilyseen that the

Lorentz transformation reallysatisfies equation (11)for cr=i ; for (11)is a consequence of (8a)and (9),and hence also of (8)and (9). We have thus derived

the Lorentz transformation.

The Lorentz transformation representedby (8)and

(9) still requiresto be generalised.Obviously it is

immaterial whether the axes of K' be chosen so that

they are spatiallyparallelto those of K. It is also not

essential that the velocityof translation of K' with

respect to K should be in the direction of the #-axis.

A simple consideration shows that we are able to

construct the Lorentz transformation in this general

sense from two kinds of transformations, viz. from

Lorentz transformations in the specialsense and from

purely spatialtransformations, which correspondsto

the replacementof the rectangularco-ordinate system

120 APPENDIX I

by a new system with its axes pointing in other

directions.

Mathematically, we can characterise the generalised

Lorentz transformation thus :

Itexpresses

x' ',y',

z', t',

in terms of linear homogeneous

functions of x, y, z, t, of such a kind that the relation

x'*+y'2+z'2-cH'z=xz+y*+z*-cH2..

(na).

is satisfied identically. That is to say : If we sub

stitute their expressions in x, y, z, t, in place of x', y',

z', t', on the left-hand side, then the left-hand side of

(na) agrees with the right-hand side.

APPENDIX II

MINKOWSKFS FOUR-

DIMENSIONAL SPACE

(" WORLD ") [SUPPLEMENTARY TO SECTION XVII]

WE can characterise the Lorentz transformation

still more simply if we introduce the imaginary

\/-

i.

ct in place of t, as time-variable. If, in

accordance with this, we insert

__

X" =J -I .Ct,

and similarly for the accented system K', then the

condition which is identically satisfied by the trans

formation can be expressed thus :

That is, by the afore-mentioned choice of "

co

ordinates," (na) is transformed into this equation.

We see from (12) that the imaginary time co-ordinate

#4 enters into the condition of transformation in exactly

the same way as the spaceco-ordinates xlt x2, x3.

It

is due to this fact that, according to the theory of

122 APPENDIX II

relativity, the " time "

x" enters into natural laws in the

same form as the space co-ordinates xv #2, *a-

A four-dimensional continuum described by the

"co-ordinates "

xlt X2, xs, x", was called " world "

by

Minkowski, who also termed a point -event a" world-

point." From a"

happening" in three-dimensional

space, physics becomes, as it were, an" existence

" in

the four-dimensional " world."

This four-dimensional " world " bears a close similarity

to the three-dimensional "

space" of (Euclidean)

analytical geometry. If we introduce into the latter a

new Cartesian co-ordinate system (x\, x'2, x'3) with

the same origin, then x'lt x'2" x'3" are linear homogeneous

functions of xlt x2, xa" which identically satisfy the

equation

%1 ~T #2 1*8 ~%l \%2 i%3"

The analogy with (12) is a complete one. We can

regard Minkowski's " world " in a formal manner as a

four-dimensional Euclidean space (with imaginary

time co-ordinate) ; the Lorentz transformation corre

sponds to a"

rotation " of the co-ordinate system in the

four-dimensional " world."

THE EXPERIMENTAL CONFIRMATION OF THE

GENERAL THEORY OF RELATIVITY

FROM a systematic theoretical point of view, we

may imagine the process of evolution of an em

pirical science to be a continuousprocess of in

duction. Theories are evolved and are expressed in

short compass as statements of a large number of in

dividual observations in the form of empirical laws,

from which the general laws can be ascertained by com

parison. Regarded in thisway, the development of a

science bears some resemblance to the compilation of a

classified catalogue. It is, as it were, a purely empirical

enterprise.

But this point of view by no means embraces the whole

of the actual process ; for it slurs over the important

part played by intuition and deductive thought in the

development of an exact science. As soon as a science

has emerged from its initial stages, theoretical advances

are no longer achieved merely by a process of arrange

ment. Guided by empirical data, the investigator

rather develops a system of thought which, in general,

is built up logically from a small number of fundamental

assumptions, the so-called axioms. We call such a

system of thought a theory. The theory finds the

124 APPENDIX III

justificationfor its existence in the fact that it correlates

a largenumber of singleobservations,and it is justhere

that the " truth " of the theorylies.

Correspondingto the same complex of empiricaldata,there may be severa] theories, which differ from one

another to a considerable extent. But as regardsthe

deductions from the theories which are capable of

being tested, the agreement between the theories may

be so complete, that it becomes difficult to find such

deductions in which the two theories differ from each

other. As an example, a case of general interest is

available in the province of biology,in the Darwinian

theory of the development of speciesby selection in

the strugglefor existence,and in the theoryof development which is based on the hypothesisof the hereditarytransmission of acquiredcharacters.

We have another instance of far-reachingagreementbetween the deductions from two theories in Newtonian

mechanics on the one hand, and the generaltheory of

relativityon the other. This agreement goes so far,

that up to the present we have been able to find only

a few deductions from the generaltheory of relativitywhich are capable of investigation,and to which the

physicsof pre-relativitydays does not also lead, and

this despitethe profound difference in the fundamental

assumptionsof the two theories. In what follows, we

shall againconsider these importantdeductions, and we

shall also discuss the empiricalevidence appertainingto

them which has hitherto been obtained.

(a) MOTION OF THE PERIHELION OF MERCURY

According to Newtonian mechanics and Newton's

law of gravitation,a planetwhich is revolvinground the

EXPERIMENTAL CONFIRMATION 125

sun would describe an ellipseround the latter,or, more

correctly,round the common centre of gravity of the

sun and the planet. In such a system, the sun, or the

common centre of gravity,lies in one of the foci of the

orbital ellipsein such a manner that,in the course of a

planet-year,the distance sun-planet grows from a

minimum to a maximum, and then decreases again to

a minimum. If instead of Newton's law we insert a

somewhat different law of attraction into the calcula

tion,we find that,accordingto this new law, the motion

would stilltake placein such a manner that the distance

sun-planet exhibits periodicvariations ; but in this

case the angle described by the line joiningsun and

planet during such a period (from perihelion" closest

proximityto the sun " to perihelion)would differ from

360". The line of the orbit would not then be a closed

one, but in the course of time it would fillup an annular

part of the orbital plane,viz. between the circle of

least and the circleof greatestdistance of the planetfrom

the sun.

According also to the generaltheory of relativity,which differs of course from the theory of Newton, a

small variation from the Newton-Kepler motion of a

planetin its orbit should take place,and in such a way,

that the angle described by the radius sun-planet

between one perihelionand the next should exceed that

correspondingto one complete revolution by an amount

givenby

(N.B." One complete revolution correspondsto the

angle2?,- in the absolute angular measure customary in

physics,and the above expressiongivesthe amount by

126 APPENDIX III

which the radius sun-planetexceeds this angle duringthe interval between one perihelionand the next.)

In this expressiona represents the major semi-axis of

the ellipse,e its eccentricity,c the velocityof light,and

T the period of revolution of the planet. Our result

may also be stated as follows : According to the general

theory of relativity,the major axis of the ellipserotates

round the sun in the same sense as the orbital motion

of the planet. Theory requiresthat this rotation should

amount to 43 seconds of arc per century for the planet

Mercury,but for the other planetsof our solar system its

magnitude should be so small that it would necessarily

escape detection.1

In point of fact, astronomers have found that the

theory of Newton does not suffice to calculate the

observed motion of Mercury with an exactness cor

respondingto that of the delicacyof observation attain

able at the present time. After taking account of all

the disturbinginfluences exerted on Mercury by the

remaining planets,it was found (Leverrier" 1859"

and Newcomb " 1895) that an unexplained perihelialmovement of the orbit of Mercury remained over, the

amount of which does not differsensiblyfrom the above-

mentioned +43 seconds of arc per century. The un

certaintyof the empiricaaresult amounts to a few

seconds only.

(b) DEFLECTION OF LIGHT BY A GRAVITATIONAL

FIELD

In Section XXII it has been alreadymentioned that,

1 Especiallysince the next planet Venus has an orbit that is

almost an exact circle,which makes it more difficult to locate

the perihelionwith precision.


according to the generaltheoryof relativity,a ray of

lightwill experiencea curvature of its path when passing

through a gravitationalfield,this curvature beingsimilar

to that experiencedby the path of a body which is

projectedthrough a gravitationalfield. As a result of

this theory,we should expect that a ray of lightwhich

is passingclose to a heavenly body would be deviated

towards the latter. For a ray of lightwhich passes the

sun at a distance of A sun-radii from its centre, the

angleof deflection (a)should amount to

i -7 seconds of arc_____

It may be added that, accordingto the theory,half of

this deflection is produced by the

Newtonian field of attraction of the i

sun, and the other half by the geo- ^D(metrical modification ("curvature") '

of space caused by the sun.,

'

This result admits of an experi- / /

mental test by means of the photo

graphic registrationof stars during

a total eclipseof the sun. The onlyD jj

reason why we must wait for a total '// 2

eclipseis because at every other /

time the atmosphere is so strongly t

illuminated by the light from the FIG. 5.

sun that the stars situated near the

sun's disc are invisible. The predictedeffect can be

seen clearlyfrom the accompanying diagram. If the

sun (S)were not present,a star which is practically

infinitelydistant would be seen in the direction Dv as

observed from the earth. But as a consequence of the

128 APPENDIX III

deflection of lightfrom the star by the sun, the star

will be seen in the direction D2, i.e. at a somewhat

greater distance from the centre of the sun than cor

respondsto itsreal position.In practice,the questionis tested in the following

way. The stars in the neighbourhood of the sun are

photographed during a solar eclipse. In addition, a

second photograph of the same stars is taken when the

sun is situated at another positionin the sky, i.e.a few

months earlier or later. As compared with the standard

photograph,the positionsof the stars on the eclipse-

photograph ought to appear displaced radiallyout

wards (away from the centre of the sun) by an amount

correspondingto the angle a.

We are indebted to the Royal Societyand to the

Royal Astronomical Society for the investigationof

this important deduction. Undaunted by the war and

by difficulties of both a material and a psychologicalnature aroused by the war, these societies equippedtwo expeditions" to Sobral (Brazil),and to the island of

Principe (West Africa)" and sent several of Britain's

most celebrated astronomers (Eddington,Cottingham,Crommelin, Davidson), in order to obtain photographsof the solar eclipseof 2Qth May, 1919. The relative

discrepanciesto be expected between the stellar photo

graphs obtained during the eclipseand the comparisonphotographs amounted to a few hundredths of a milli

metre only. Thus great accuracy was necessary in

making the adjustmentsrequiredfor the taking of the

photographs,and in their subsequentmeasurement.The results of the measurements confirmed the theory

in a thoroughlysatisfactorymanner. The rectangular

components of the observed and of the calculated


deviations of the stars (in seconds of arc) are set forth

in the followingtable of results :

(c)DISPLACEMENT OF SPECTRAL LINES TOWARDS

THE RED

In Section XXIII it has been shown that in a system K'

which is in rotation with regard to a Galileian system K,

clocks of identical construction, and which are con

sidered at rest with respect to the rotating reference-

body, go at rates which are dependent on the positions

of the clocks. We shall now examine this dependence

quantitatively. A dock, which is situated at a distance

r from the centre of the disc, has a velocityrelative to

K which is given by

where w represents the angular velocityof rotation of the

disc K' with respect to K, If r0 represents the number

of ticks of the clock per unit time (" rate" of the clock)

relative to K when the clock is at rest, then the "

rate"

of the clock (v)when it is moving relative to K with

a velocityv, but at rest with respect to the disc, will,

in accordance with Section XII, be given by

130 APPENDIX III

or with sufficient accuracy by

This expressionmay also be stated in the followingform :

I 0)V2

If we represent the difference of potentialof the centri

fugal force between the positionof the clock and the

centre of the disc by "",i.e. the work, considered nega

tively,which must be performed on the unit of mass

against the centrifugalforce in order to transport it

from the positionof the clock on the rotatingdisc to

the centre of the disc,then we have

From this it follows that

.'In the firstplace,we see from this expressionthat two

' clocks of identical construction will go at different rates

when situated at different distances from the centre of

* the disc. This result is also valid from the standpointof an observer who is rotatingwith the disc.

Now, as judged from the disc,the latter is in a gravitational field of potential"",hence the result we have

obtained will hold quite generallyfor gravitational

fields. Furthermore, we can regard an atom which is

emittingspectrallines as a clock,so that the following

statement will hold :

An atom absorbs or emits lightof a frequencywhich is


dependent on the potentialof the gravitationalfieldinwhich it is situated.

-y The frequencyot an atom situated on the surface of a

heavenlybody will be somewhat less than the frequencyof an atom of the same element which is situated in free

space (or on the surface of a smaller celestial body).M

Now ""= -K-, where K is Newton's constant of

gravitation,and M is the mass of the heavenlybody.Thus a displacementtowards the red ought to take placefor spectrallines produced at the surface of stars as

compared with the spectrallines of the same element

produced at the surface of the earth,the amount of this

displacementbeing

For the sun, the displacement towards the red pre

dicted by theory amounts to about two millionths of

the wave-length. A trustworthy calculation is not

possiblein the case of the stars, because in generalneither the mass M nor the radius r is known.

It is an open questionwhether or not this effect

exists,and at the present time astronomers are workingwith great zeal towards the solution. Owing to the

smallness of the effect in the case of the sun, it is diffi

cult to form an opinion as to its existence. Whereas

Grebe and Bachem (Bonn), as a result of their own

measurements and those of Evershed and Schwarzschild

on the cyanogen bands, have placedthe existence of

the effect almost beyond doubt, other investigators,

particularlySt. John, have been led to the opposite

opinionin consequence of their measurements.

132 APPENDIX in

Mean displacements of lines towards the less re

frangible end of the spectrum are certainly revealed by

statistical investigations of the fixed stars ; but up

to the present the examination of the available data

does not allow of anydefinite decision being arrived at,

as to whether or not these displacements are to be

referred in reality to the effect of gravitation. The

results of observation have been collected together,

and discussed in detail from the standpoint of the

question which has been engaging our attention here,

in a paper by E. Freundlich entitled " Zur Priifung der

aligemeinen Relativitats-Theorie "

(Die Naturwissen-

schaften, 1919, No. 35, p. 520 : Julius Springer, Berlin).

At all events, a definite decision will be reached during

the next few years. If the displacement of spectral

lines towards the red by the gravitational potential

does not exist, then the general theory of relativity

will be untenable. On the other hand, if the cause of

the displacement of spectral lines be definitely traced

to the gravitational potential, then the study of this

displacement will furnish us with important informa

tion as to the mass of the heavenly bodies.

BIBLIOGRAPHY

WORKS IN ENGLISH ON EINSTEIN'S THEORY

INTRODUCTORY

The Foundations of Einstein's Theory of Gravitation:

Erwin Freundlich (translation by H. L. Brose).Carab. Univ. Press, 1920.

Space and Time in Contemporary Physics \Moritz Schlick

(translation by H. L. Brose). Clarendon Press,

Oxford, 1920.

THE SPECIAL THEORY

The Principle of Relativity :E. Cunningham. Catnb.

Univ. Press.

Relativity and the Electron Theory : E. Cunningham, Mono

graphs on Physics. Longmans, Green " Co.

The Theory of Relativity :L. Silberstein. Macmillan " Co.

The Space-Time Manifold of Relativity -.E. B. Wilson

and G. N. Lewis, Proc. Amer. Soc. Arts "S- Science.

vol. xlviii., No. n, 1912.

THE GENERAL THEORY

Report on the Relativity Theory of Gravitation :A. S.

Eddington. Fleetway Press Ltd., Fleet Street,

London.

]84 BIBLIOGRAPHY

On Einstein's Theory of Gravitation and its Astronomical

Consequences : W. de Sitter, M. N. Roy. Astron.

Soc., Ixxvi. p. 699, 1916 ;Ixxvii.

p. 155, 1916 ;Ixxviii.

P- 3.

On Einstein's Theory of Gravitation:

H. A. Lorentz, Proc.

Amsterdam Acad., vol. xix.p. 1341, 1917.

Space, Time and Gravitation:

W. de Sitter: The

Observatory, No. 505, p. 412. Taylor " Francis, Fleet

Street, London.

The Total Eclipse of 2Qth May, 1919, and the Influence of

Gravitation on Light : A. S. Eddington, ibid.,

March 1919.

Discussion on the Theory of Relativity :M. N. Roy. Astron.

Soc., vol. Ixxx. No. 2, p. 96, December 1919.

The Displacement of Spectrum Lines and the Equivalence

Hypothesis :W. G. Duffield, M. N. Roy. Astron. Soc.,

vol. Ixxx.;

No. 3, p. 262, 1920.

Space, Time and Gravitation:

A. S. Eddington, Camb. Univ.

Press, 1920.

ALSO, CHAPTERS IN

The Mathematical Theory of Electricity and Magnetism :

J. H. Jeans (4th edition). Camb. Univ. Press, 1920.

The Electron Theory of Matter:

O. W. Richardson. Camb.

Univ. Press. "

INT)EX

Aberration, 49

Absorption of energy, 46

Acceleration, 64, 67, 70

Action at a distance, 48

Addition of velocities, 16, 38

Adjacent points, 89

Aether, 52

drift, 52, 53

Arbitrary substitutions, 98

Astronomy, 7, 102

Astronomical day, n

Axioms, 2, 123

"truth of, 2

Bachem, 131

Basis of theory, 44

" Being," 66, 108

^-rays, 50

Biology, 124

Cartesian system of co-ordi

nates, 7, 84, 122

Cathode rays, 50

Celestial mechanics, 105

Centrifugal force, 80, 130

Chest, 66

Classical mechanics, 9, 13, 14,

16, 30, 44, 71, 102, 103, 124

truth of, 13

Clocks, 10, 23, 80, 81, 94, 95,

98-100, 102, 113, 129

"rate of, 129

Conception of mass, 45

" position, 6

Conservation of energy, 45, 101

" impulse, 101

" mass, 45, 47

Continuity, 95

Continuum, 55, 83

" two-dimensional, 94

" three-dimensional, 57

" four-dimensional, 89, 91, 92

94, 1*2

" space-time, 78, 91-96

" Euclidean, 84, 86, 88, 92

" non- Euclidean, 86, 90

Co-ordinate differences, 92

" differentials, 92

" planes, 32

Cottingham, 128

Counter- Point, 112

Co- variant, 43

Crommelin, 128

Curvature of light-rays, 104, \zy

" space, 127

Curvilinear motion, 74

Cyanogen bands, 131

Darwinian theory, 1 24

Davidson, 128

Deductive thought, 123

Derivation of laws, 44

De Sitter, 17

Displacement of spectral lines,

104, 129

Distance (line-interval), 3, 5 8,

28, 29, 84, 88, 109

" physical interpretation of, 5

" relativity of, 28

Doppler principle, 50.

Double stars, 17

Eclipse of star, 17

Eddington, 104, 128

136 INDEX

Electricity,76Electrodynamics,13, 19, 41, 44,

76Electromagnetic theory,49

" waves, 63Electron, 44, 50." electrical masses of,5 1

Electrostatics,76Ellipticalspace, 112

Empiricallaws, 123

Encounter (space-time coin

cidence),95Equivalent,14Euclidean geometry, i, 2, 57,

82, 86, 88, 108, 109, 113,122

propositionsof,3,8" space, 57, 86, 122

Evershed, 131

Experience,49, 60

Faraday, 48, 63FitzGerald, 53Fixed stars, 1 1

Fizeau, 39, 49, 5 1

" experiment of, 39

Frequency of atom, 131

Galilei,n" transformation,33, 36, 38, 42,

52

Galileian system of co-ordinates,ii, 13, 14, 46, 79, 91, 98,100

Gauss, 86, 87, 90

Gaussian co-ordinates,88-90, 94,

96-100General theory of relativity,

59-104, 97Geometrical ideas,2, 3" propositions,i

truth of,2-4Gravitation, 64, 69, 78, 102

Gravitational field,64, 67, 74,

77. 93, 98, 100, 101, 113

potentialof, 130, 131

" mass, 65, 68, 102

Grebe, 131

Group-densityof stars, 106

Helmholtz, 108

Heuristic value of relativity,42

Induction, 123

Inertia,65Inertial mass, 47, 65, 69, 101,

102

Instantaneous photograph(snapshot),117

Intensityof gravitationalfield,106

Intuition,123Ions, 44

Kepler,125Kinetic energy, 45, 101

Lattice,108

Law of inertia,n, 61, 62, 98Laws of Galilei-Newton, 1 3

" of Nature, 60, 71, 99

Leverrier, 103, 126

Light-signal,33, 115, 118

Light-stimulus,33Limiting velocity(c),36, 37Lines of force, 106

Lorentz, H. A., 19, 41, 44, 49,

50-3" transformation, 33, 39, 42,

91, 97, 98, 115, 118, 119,

121

(generalised),120

Mach, E., 72

Magnetic field,63Manifold (seeContinuum)Mass of heavenly bodies, 132

Matter, 101

Maxwell, 41, 44, 48-50, 52" fundamental equations, 46,

77Measurement of length,85Measuring-rod, 5, 6, 28, 80, 81,

94, loo, 102, in, 113,

117

Mercury, 103, 126

" orbit of, 103, 126

Michelson, 52-4

Minkowski, 55-57, 91, 122

INDEX 137

Morley,53, 54

Motion, 14, 60

" of heavenly bodies, 13, 15,

44, 102, 113

New comb, 126

Newton, n, 72, 102, 105, 125

Newton's constant of gravitation,131

" law of gravitation, 48, 80,106, 124

" law of motion, 64Non-Euclidean geometry, 108

Non-Galileian reference-bodies,98.

Non-uniform motion, 62

Optics,13, 19, 44

Organ-pipe,note of, 14

Parabola, 9, 10

Path-curve, 10

Perihelion of Mercury, 124-126

Physics,7"

"ofmeasurement, 7

Place specification,5, 6

Plane, i, 108, 109

Poincar6, 108

Point, i

Point-mass, energy of,45

Position, 9

Principleof relativity,13-15,19, 20, 60

Processes of Nature, 42

Propagation of light, 17, 19,

20, 32, 91, 119

in liquid,40in gravitationalfields,75

Quasi-Euclidean universe, 114

Quasi-sphericaluniverse, 114

Radiation, 46Radioactive substances, 50

Reference-body, 5, 7, 9-11, 18,

23, 25, 26, 37, 60

rotating,79" mollusk, 99-101

Relative position,-3" velocity,1 1 7

Rest, 14

Riemann, 86, 108, in

Rotation, 81, 122

Schwarzschild,131Seconds- clock, 36Seeliger,105, 106

Simultaneity,22, 24-26, 81

" relativityof,26Size-relations,90Solar eclipse,75, 127, 128

Space,9, 52, 55, 105" conception of, 19

Space co-ordinates, 55, 81, 99

Space-interval,30, 56" point,99

" two-dimensional, 108

" three-dimensional,122

Special theory of relativity,i-S7, 20

Sphericalsurface, 109

" space, in, 112.

St. John, 131

Stellar universe, 106

" photographs, 128

Straightline, 1-3, 9, 10, 82, 88,

109

System of co-ordinates,5, 10, 1 1

Terrestrial space, 15

Theory, 123

" truth of, 1 24

Three-dimensional, 55

Time, conception of, 19, 52,

105

" co-ordinate, 55, 99

" in Physics,21, 98, 122

" of an event, 24, 26

Time-interval, 30, 56

Trajectory, 10

" Truth," 2

Uniform translation,12, 59

Universe (World) structure of,

108, 113

" circumference of, 1 1 1

138 INDEX

Universe, elliptical, 112, 114

"

Euclidean, 109, 1 1 1

" space expanse (radius) of,

114

"

spherical, 1 1 1, 114

Value ofTT,

82, no

Velocity of light, 10, 17, 18, 76,

118

Venus, 126

Weight (heaviness), 65

World, 55, 56, 109, 122

World-point, 122

radius, 112

sphere, no, in

Zeeman, 41

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